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<?xml version="1.0" encoding="utf-8"?>
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<title>linkfqy</title>
<subtitle>A link to the past.</subtitle>
<link href="/atom.xml" rel="self"/>
<link href="http://oi.linkfqy.top/"/>
<updated>2021-08-24T13:10:08.396Z</updated>
<id>http://oi.linkfqy.top/</id>
<author>
<name>linkfqy</name>
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<generator uri="http://hexo.io/">Hexo</generator>
<entry>
<title>数模国赛2020A题</title>
<link href="http://oi.linkfqy.top/posts/%E6%95%B0%E6%A8%A1%E5%9B%BD%E8%B5%9B2020A%E9%A2%98/"/>
<id>http://oi.linkfqy.top/posts/数模国赛2020A题/</id>
<published>2021-08-24T12:32:58.000Z</published>
<updated>2021-08-24T13:10:08.396Z</updated>
<content type="html"><![CDATA[<h1 id="第一问">第一问</h1><p>忽略电路板厚度，假定温区内部恒温</p><p>通过热传导方程得到不同温区之间的空隙中，温度在空间上线性分布</p><p>通过牛顿冷却定律和比热容公式解微分方程可知，物体在恒温环境下温度随时间的函数</p><p>然后由附件数据分段拟合参数</p><p>下面是拟合的代码</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">import</span> pandas <span class="keyword">as</span> pd</span><br><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> curve_fit</span><br><span class="line">exc = pd.read_excel(<span class="string">"input.xlsx"</span>, usecols=range(<span class="number">0</span>, <span class="number">2</span>))</span><br><span class="line">data = exc.values</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">pos = np.array([[<span class="number">0</span>, <span class="number">0</span>], [<span class="number">25</span>, <span class="number">55.5</span>], [<span class="number">60.5</span>, <span class="number">91</span>], [<span class="number">96</span>, <span class="number">126.5</span>], [<span class="number">131.5</span>, <span class="number">162</span>], [<span class="number">167</span>, <span class="number">197.5</span>], [<span class="number">202.5</span>, <span class="number">233</span>], [<span class="number">238</span>, <span class="number">268.5</span>], [<span class="number">273.5</span>, <span class="number">304</span>], [<span class="number">309</span>, <span class="number">339.5</span>], [<span class="number">344.5</span>, <span class="number">375</span>], [<span class="number">380</span>, <span class="number">410.5</span>], [<span class="number">435.5</span>, <span class="number">460.5</span>]]</span><br><span class="line"> )</span><br><span class="line"></span><br><span class="line">ranx = np.array([[<span class="number">0</span>, pos[<span class="number">2</span>][<span class="number">0</span>]], [pos[<span class="number">2</span>][<span class="number">0</span>], pos[<span class="number">5</span>][<span class="number">1</span>]], [pos[<span class="number">5</span>][<span class="number">1</span>], pos[<span class="number">6</span>][<span class="number">0</span>]],</span><br><span class="line"> [pos[<span class="number">6</span>][<span class="number">0</span>], pos[<span class="number">6</span>][<span class="number">1</span>]], [pos[<span class="number">6</span>][<span class="number">1</span>], pos[<span class="number">7</span>][<span class="number">0</span>]], [</span><br><span class="line"> pos[<span class="number">7</span>][<span class="number">0</span>], pos[<span class="number">7</span>][<span class="number">1</span>]],</span><br><span class="line"> [pos[<span class="number">7</span>][<span class="number">1</span>], pos[<span class="number">8</span>][<span class="number">0</span>]], [pos[<span class="number">8</span>][<span class="number">0</span>], pos[<span class="number">9</span>][<span class="number">1</span>]], [</span><br><span class="line"> pos[<span class="number">9</span>][<span class="number">1</span>], pos[<span class="number">10</span>][<span class="number">1</span>]],</span><br><span class="line"> [pos[<span class="number">10</span>][<span class="number">1</span>], pos[<span class="number">12</span>][<span class="number">0</span>]]])</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">getab</span><span class="params">(i, tem)</span>:</span> <span class="comment"># 计算第i个小温区后的温度曲线T=ax+b</span></span><br><span class="line"> x1 = pos[i][<span class="number">1</span>]</span><br><span class="line"> x2 = pos[i+<span class="number">1</span>][<span class="number">0</span>]</span><br><span class="line"> t1 = tem[i]</span><br><span class="line"> t2 = tem[i+<span class="number">1</span>]</span><br><span class="line"> a = (t2-t1)/(x2-x1)</span><br><span class="line"> b = t1-a*x1</span><br><span class="line"> <span class="keyword">return</span> (a, b)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">getfx</span><span class="params">(x, tem)</span>:</span> <span class="comment"># 获取x位置的环境温度曲线f(x)=ax+b</span></span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> range(<span class="number">0</span>, <span class="number">12</span>):</span><br><span class="line"> <span class="keyword">if</span> (pos[i][<span class="number">0</span>] <= x <span class="keyword">and</span> x <= pos[i][<span class="number">1</span>]):</span><br><span class="line"> <span class="keyword">return</span> (<span class="number">0</span>, tem[i])</span><br><span class="line"> <span class="keyword">elif</span> (pos[i][<span class="number">1</span>] <= x <span class="keyword">and</span> x <= pos[i+<span class="number">1</span>][<span class="number">0</span>]):</span><br><span class="line"> <span class="keyword">return</span> getab(i, tem=tem)</span><br><span class="line"> <span class="keyword">return</span> (<span class="number">-1</span>, <span class="number">-1</span>)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(tt, lmd)</span>:</span> <span class="comment"># 用于拟合的函数</span></span><br><span class="line"> <span class="function"><span class="keyword">def</span> <span class="title">gettem</span><span class="params">(t)</span>:</span> <span class="comment"># 获取x时刻的零件温度，已知初值T(t0)=T0</span></span><br><span class="line"> <span class="comment"># a, b = (0, 175)</span></span><br><span class="line"> a, b = getfx(t[<span class="number">0</span>]*v, tem=tem)</span><br><span class="line"> c = (T0-a*v*t0+a*v/lmd-b)*np.exp(lmd*t0)</span><br><span class="line"> <span class="keyword">return</span> a*v*t-a*v/lmd+b+c*np.exp(-lmd*t)</span><br><span class="line"> <span class="keyword">return</span> gettem(tt)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line"> tem = np.array([<span class="number">25</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>,</span><br><span class="line"> <span class="number">198</span>, <span class="number">230</span>, <span class="number">257</span>, <span class="number">257</span>, <span class="number">25</span>, <span class="number">25</span>, <span class="number">25</span>])</span><br><span class="line"> v = <span class="number">70</span>/<span class="number">60</span> <span class="comment"># 传送带速度，单位cm/s</span></span><br><span class="line"></span><br><span class="line"> tT = data</span><br><span class="line"> t = tT.T[<span class="number">0</span>]</span><br><span class="line"> T = tT.T[<span class="number">1</span>]</span><br><span class="line"> lmds = []</span><br><span class="line"> TT = []</span><br><span class="line"> x = t*v</span><br><span class="line"> lj, j = <span class="number">0</span>, <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> ranxi <span class="keyword">in</span> ranx:</span><br><span class="line"> <span class="keyword">while</span> (j < len(x) <span class="keyword">and</span> x[j] <= ranxi[<span class="number">1</span>]):</span><br><span class="line"> j += <span class="number">1</span></span><br><span class="line"> t0 = t[lj]</span><br><span class="line"> T0 = T[lj]</span><br><span class="line"> popt, pcov = curve_fit(f, t[lj:j], T[lj:j])</span><br><span class="line"> lmds.extend(popt)</span><br><span class="line"> TT.extend(f(t[lj:j], lmd=popt))</span><br><span class="line"> lj = j</span><br><span class="line"> <span class="keyword">if</span> (j >= len(x)):</span><br><span class="line"> <span class="keyword">break</span></span><br><span class="line"></span><br><span class="line"> plt.scatter(t, T, s=<span class="number">1</span>)</span><br><span class="line"> plt.plot(t, TT, color=<span class="string">'r'</span>)</span><br><span class="line"> plt.savefig(<span class="string">'getlmd.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line"> plt.show()</span><br><span class="line"> print(lmds)</span><br></pre></td></tr></table></figure><p>第一问代码：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">import</span> pandas <span class="keyword">as</span> pd</span><br><span class="line"><span class="keyword">from</span> getlmd <span class="keyword">import</span> ranx, pos, getfx</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment"># lmd = 0.01442337549312217</span></span><br><span class="line">lmds = [<span class="number">0.008643609716781697</span>, <span class="number">0.02103327687676206</span>, <span class="number">0.010922351732360641</span>, <span class="number">0.014168584576603811</span>, <span class="number">0.01604373036324419</span>,</span><br><span class="line"> <span class="number">0.0269012209873633</span>, <span class="number">0.023523032208845447</span>, <span class="number">0.0190525180588788</span>, <span class="number">0.00079443137454158</span>, <span class="number">0.009429030064290005</span>]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">gettem</span><span class="params">(t, t0, T0, lmd, v, tem)</span>:</span> <span class="comment"># 获取t时段的零件温度，已知初值T(t0)=T0</span></span><br><span class="line"> a, b = getfx(t[<span class="number">0</span>]*v, tem=tem)</span><br><span class="line"> c = (T0-a*v*t0+a*v/lmd-b)*np.exp(lmd*t0)</span><br><span class="line"> <span class="keyword">return</span> a*v*t-a*v/lmd+b+c*np.exp(-lmd*t)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">gettT</span><span class="params">(v, tem, dt=<span class="number">0.5</span>, fil=lambda x: x[<span class="number">1</span>] >= <span class="number">30</span>)</span>:</span> <span class="comment"># 计算温度曲线</span></span><br><span class="line"> t = np.arange(dt, pos[<span class="number">-1</span>][<span class="number">0</span>]/v, dt)</span><br><span class="line"> T = []</span><br><span class="line"> x = t*v</span><br><span class="line"> lj, j = <span class="number">0</span>, <span class="number">0</span></span><br><span class="line"> t0, T0 = <span class="number">0</span>, <span class="number">25</span></span><br><span class="line"> <span class="keyword">for</span> (i, ranxi) <span class="keyword">in</span> enumerate(ranx):</span><br><span class="line"> <span class="keyword">while</span> (j < len(x) <span class="keyword">and</span> x[j] <= ranxi[<span class="number">1</span>]):</span><br><span class="line"> j += <span class="number">1</span></span><br><span class="line"> Ti = gettem(t[lj:j], t0, T0, lmds[i], v=v, tem=tem)</span><br><span class="line"> T.extend(Ti)</span><br><span class="line"> lj = j</span><br><span class="line"> t0, T0 = t[j<span class="number">-1</span>], T[j<span class="number">-1</span>]</span><br><span class="line"> <span class="keyword">if</span> (j >= len(x)):</span><br><span class="line"> <span class="keyword">break</span></span><br><span class="line"></span><br><span class="line"> tT = np.array([t, T]).T</span><br><span class="line"> <span class="keyword">if</span> (fil != <span class="keyword">None</span>):</span><br><span class="line"> tT_f = filter(fil, tT)</span><br><span class="line"> tT = np.array(list(tT_f))</span><br><span class="line"> t, T = tT.T</span><br><span class="line"> <span class="keyword">return</span> (t, T, tT)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line"> tem = np.array([<span class="number">25</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>, <span class="number">173</span>,</span><br><span class="line"> <span class="number">198</span>, <span class="number">230</span>, <span class="number">257</span>, <span class="number">257</span>, <span class="number">25</span>, <span class="number">25</span>, <span class="number">25</span>])</span><br><span class="line"> v = <span class="number">78</span>/<span class="number">60</span> <span class="comment"># 传送带速度，单位cm/s</span></span><br><span class="line"></span><br><span class="line"> t, T, tT = gettT(v=v, tem=tem)</span><br><span class="line"> plt.plot(t, T)</span><br><span class="line"> plt.savefig(<span class="string">'sub1.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line"> plt.show()</span><br><span class="line"> res = pd.DataFrame(tT, columns=[<span class="string">"时间"</span>, <span class="string">"温度"</span>])</span><br><span class="line"> res.to_csv(<span class="string">'result.csv'</span>, index=<span class="keyword">False</span>, header=<span class="keyword">False</span>)</span><br><span class="line"> printtim = [(pos[<span class="number">3</span>][<span class="number">0</span>]+pos[<span class="number">3</span>][<span class="number">1</span>])/<span class="number">2</span>/v, (pos[<span class="number">6</span>][<span class="number">0</span>]+pos[<span class="number">6</span>][<span class="number">1</span>]) /</span><br><span class="line"> <span class="number">2</span>/v, (pos[<span class="number">7</span>][<span class="number">0</span>]+pos[<span class="number">7</span>][<span class="number">1</span>])/<span class="number">2</span>/v, pos[<span class="number">8</span>][<span class="number">1</span>]/v]</span><br><span class="line"> lmdid = [<span class="number">1</span>, <span class="number">3</span>, <span class="number">5</span>, <span class="number">7</span>]</span><br><span class="line"> j = <span class="number">1</span></span><br><span class="line"> <span class="keyword">for</span> (i, timei) <span class="keyword">in</span> enumerate(printtim):</span><br><span class="line"> <span class="keyword">while</span> (j < len(t) <span class="keyword">and</span> t[j] < timei):</span><br><span class="line"> j += <span class="number">1</span></span><br><span class="line"> <span class="keyword">if</span> j >= len(t):</span><br><span class="line"> <span class="keyword">break</span></span><br><span class="line"> print(gettem(np.array([timei]), t[j<span class="number">-1</span>], T[j<span class="number">-1</span>],</span><br><span class="line"> lmd=lmds[lmdid[i]], v=v, tem=tem))</span><br></pre></td></tr></table></figure><p>得到的各点温度：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">[135.71310782]</span><br><span class="line">[169.75603785]</span><br><span class="line">[191.00134962]</span><br><span class="line">[223.82106867]</span><br></pre></td></tr></table></figure><p>炉温曲线：</p><p><img src="/posts/数模国赛2020A题/sub1.png"></p><h1 id="第二问">第二问</h1><p>大力枚举速度v，先大范围粗糙枚举，再小范围精确枚举</p><p>需要复用许多第一问的代码，所以函数化编程很重要，同时也要封装好来</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">from</span> sub1 <span class="keyword">import</span> gettT</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">checkv</span><span class="params">(t, T, tT, dt=<span class="number">0.5</span>)</span>:</span></span><br><span class="line"> maxT = np.max(T)</span><br><span class="line"> <span class="keyword">if</span> <span class="keyword">not</span>(<span class="number">240</span> <= maxT <span class="keyword">and</span> maxT <= <span class="number">250</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">False</span></span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> range(<span class="number">1</span>, len(T)):</span><br><span class="line"> <span class="keyword">if</span> np.abs((T[i]-T[i<span class="number">-1</span>])/dt) > <span class="number">3</span>:</span><br><span class="line"> print((i, T[i], T[i<span class="number">-1</span>], t[i], t[i<span class="number">-1</span>]))</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">False</span></span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">150</span> <span class="keyword">and</span> T[i] >= <span class="number">150</span>):</span><br><span class="line"> t1 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">190</span> <span class="keyword">and</span> T[i] >= <span class="number">190</span>):</span><br><span class="line"> t2 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">217</span> <span class="keyword">and</span> T[i] >= <span class="number">217</span>):</span><br><span class="line"> t3 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] >= <span class="number">217</span> <span class="keyword">and</span> T[i] < <span class="number">217</span>):</span><br><span class="line"> t4 = t[i]</span><br><span class="line"> <span class="keyword">if</span> <span class="keyword">not</span> (<span class="number">60</span> <= t2-t1 <span class="keyword">and</span> t2-t1 <= <span class="number">120</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">False</span></span><br><span class="line"> <span class="keyword">if</span> <span class="keyword">not</span> (<span class="number">40</span> <= t4-t3 <span class="keyword">and</span> t4-t3 <= <span class="number">90</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">False</span></span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">True</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">printlim</span><span class="params">(t, T, tT, dt=<span class="number">0.5</span>)</span>:</span> <span class="comment"># 输出各制程的极限值</span></span><br><span class="line"> maxT = np.max(T)</span><br><span class="line"> print(<span class="string">"maxT="</span>, maxT)</span><br><span class="line"> maxk = <span class="number">0</span></span><br><span class="line"> mink = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> range(<span class="number">1</span>, len(T)):</span><br><span class="line"> maxk = max((T[i]-T[i<span class="number">-1</span>])/dt, maxk)</span><br><span class="line"> mink = min((T[i]-T[i<span class="number">-1</span>])/dt, mink)</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">150</span> <span class="keyword">and</span> T[i] >= <span class="number">150</span>):</span><br><span class="line"> t1 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">190</span> <span class="keyword">and</span> T[i] >= <span class="number">190</span>):</span><br><span class="line"> t2 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] < <span class="number">217</span> <span class="keyword">and</span> T[i] >= <span class="number">217</span>):</span><br><span class="line"> t3 = t[i]</span><br><span class="line"> <span class="keyword">if</span> (T[i<span class="number">-1</span>] >= <span class="number">217</span> <span class="keyword">and</span> T[i] < <span class="number">217</span>):</span><br><span class="line"> t4 = t[i]</span><br><span class="line"> print(<span class="string">"maxk="</span>, maxk)</span><br><span class="line"> print(<span class="string">"mink="</span>, mink)</span><br><span class="line"> print(<span class="string">"t2-t1="</span>, t2-t1)</span><br><span class="line"> print(<span class="string">"t4-t3="</span>, t4-t3)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line"> tem = np.array([<span class="number">25</span>, <span class="number">182</span>, <span class="number">182</span>, <span class="number">182</span>, <span class="number">182</span>, <span class="number">182</span>,</span><br><span class="line"> <span class="number">203</span>, <span class="number">237</span>, <span class="number">254</span>, <span class="number">254</span>, <span class="number">25</span>, <span class="number">25</span>, <span class="number">25</span>])</span><br><span class="line"> maxv = <span class="number">0</span></span><br><span class="line"> dt = <span class="number">0.1</span></span><br><span class="line"> <span class="keyword">for</span> v <span class="keyword">in</span> np.linspace(<span class="number">65</span>/<span class="number">60</span>, <span class="number">100</span>/<span class="number">60</span>, num=<span class="number">35</span>*<span class="number">2</span>+<span class="number">1</span>): <span class="comment"># 间隔0.5</span></span><br><span class="line"> t, T, tT = gettT(v=v, dt=dt, tem=tem)</span><br><span class="line"> <span class="comment"># print("v={}".format(v))</span></span><br><span class="line"> <span class="keyword">if</span> (checkv(t, T, tT, dt)):</span><br><span class="line"> maxv = v</span><br><span class="line"> print(maxv*<span class="number">60</span>)</span><br><span class="line"> <span class="keyword">for</span> v <span class="keyword">in</span> np.linspace(<span class="number">70</span>/<span class="number">60</span>, <span class="number">75</span>/<span class="number">60</span>, num=<span class="number">5</span>*<span class="number">100</span>+<span class="number">1</span>): <span class="comment"># 间隔0.01</span></span><br><span class="line"> t, T, tT = gettT(v=v, dt=dt, tem=tem)</span><br><span class="line"> <span class="comment"># print("v={}".format(v))</span></span><br><span class="line"> <span class="keyword">if</span> (checkv(t, T, tT, dt)):</span><br><span class="line"> maxv = v</span><br><span class="line"> print(maxv*<span class="number">60</span>)</span><br><span class="line"> t, T, tT = gettT(v=maxv, dt=dt, tem=tem)</span><br><span class="line"> printlim(t, T, tT, dt=dt)</span><br><span class="line"> plt.plot(t, T)</span><br><span class="line"> plt.savefig(<span class="string">'sub2.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line"> plt.show()</span><br></pre></td></tr></table></figure><p>得到的各参数：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">v1=72.0</span><br><span class="line">v2=72.33</span><br><span class="line">maxT= 240.00007138887304</span><br><span class="line">maxk= 2.658684542907963</span><br><span class="line">mink= -1.8523553812633509</span><br><span class="line">t2-t1= 103.69999999999999</span><br><span class="line">t4-t3= 82.90000000000006</span><br></pre></td></tr></table></figure><p>在最大速度下的炉温曲线：</p><p><img src="/posts/数模国赛2020A题/sub2.png"></p><h1 id="第三问">第三问</h1><p>把阴影面积作为估值函数，做模拟退火</p><p>调了<code>scikit-opt</code>的库</p><p>然后还画了最小面积随迭代次数变化的曲线</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sko.SA <span class="keyword">import</span> SA, SABoltzmann, SAFast</span><br><span class="line"><span class="keyword">from</span> sub1 <span class="keyword">import</span> gettT</span><br><span class="line"><span class="keyword">from</span> sub2 <span class="keyword">import</span> checkv</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> pandas <span class="keyword">as</span> pd</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line">exc = pd.read_excel(<span class="string">"input.xlsx"</span>, usecols=range(<span class="number">0</span>, <span class="number">2</span>))</span><br><span class="line">data = exc.values</span><br><span class="line">data = data.T</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">getarea</span><span class="params">(dt, T)</span>:</span></span><br><span class="line"> res = <span class="number">0</span></span><br><span class="line"> maxT = max(T)</span><br><span class="line"> id = T.tolist().index(maxT)</span><br><span class="line"> <span class="keyword">for</span> (i, Ti) <span class="keyword">in</span> enumerate(T):</span><br><span class="line"> <span class="keyword">if</span> (Ti >= <span class="number">217</span> <span class="keyword">and</span> i <= id):</span><br><span class="line"> res += dt*(Ti<span class="number">-217</span>)</span><br><span class="line"> <span class="keyword">return</span> res</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">X2v</span><span class="params">(X)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> (X[<span class="number">0</span>]*<span class="number">35</span>/<span class="number">20</span>+<span class="number">65</span>)/<span class="number">60</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">X2tem</span><span class="params">(X)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> [<span class="number">25</span>]+[<span class="number">175</span>+X[<span class="number">1</span>]<span class="number">-10</span>]*<span class="number">5</span>+[<span class="number">195</span>+X[<span class="number">2</span>]<span class="number">-10</span>] + \</span><br><span class="line"> [<span class="number">235</span>+X[<span class="number">3</span>]<span class="number">-10</span>]+[<span class="number">255</span>+X[<span class="number">4</span>]<span class="number">-10</span>]*<span class="number">2</span>+[<span class="number">25</span>]*<span class="number">3</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">cost</span><span class="params">(X)</span>:</span> <span class="comment"># 计算模拟退火的代价函数即阴影面积</span></span><br><span class="line"> <span class="keyword">if</span> (X.min() < <span class="number">0</span> <span class="keyword">or</span> X.max() > <span class="number">20</span>): <span class="comment"># 判断输入向量是否合法</span></span><br><span class="line"> <span class="keyword">return</span> <span class="number">1e10</span>+np.abs(X.min()<span class="number">-0</span>)+np.abs(X.max()<span class="number">-20</span>)</span><br><span class="line"> v = X2v(X)</span><br><span class="line"> tem = X2tem(X)</span><br><span class="line"> t, T, tT = gettT(v=v, tem=tem, dt=<span class="number">0.1</span>)</span><br><span class="line"> <span class="keyword">if</span> (<span class="keyword">not</span> checkv(t, T, tT, dt=<span class="number">0.1</span>)): <span class="comment"># 判断是否满足制程界限</span></span><br><span class="line"> <span class="keyword">return</span> <span class="number">1e5</span></span><br><span class="line"> <span class="keyword">return</span> getarea(dt=<span class="number">0.1</span>, T=T)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line"> sa = SA(func=cost, x0=[<span class="number">0</span>]*<span class="number">5</span>, lb=<span class="number">0</span>, ub=<span class="number">20</span>, max_stay_counter=<span class="number">5000</span>, L=<span class="number">300</span>)</span><br><span class="line"> best_x, best_y = sa.run()</span><br><span class="line"> print(<span class="string">"best_x={}\nminArea={}"</span>.format(best_x, best_y))</span><br><span class="line"> v = X2v(best_x)</span><br><span class="line"> tem = X2tem(best_x)</span><br><span class="line"> print(<span class="string">"v={}\ntem={}\n"</span>.format(v*<span class="number">60</span>, tem))</span><br><span class="line"> t, T, tT = gettT(v=v, tem=tem, dt=<span class="number">0.1</span>)</span><br><span class="line"> plt.plot(t, T)</span><br><span class="line"> plt.show()</span><br><span class="line"> by_f = filter(<span class="keyword">lambda</span> x: x < <span class="number">1e3</span>, sa.best_y_history)</span><br><span class="line"> by = list(by_f)</span><br><span class="line"> plt.plot(pd.DataFrame(by).cummin(axis=<span class="number">0</span>)) <span class="comment"># 画出最小面积随迭代次数变化的曲线</span></span><br><span class="line"> plt.show()</span><br><span class="line"> plt.savefig(<span class="string">'sub3_2.png'</span>, dpi=<span class="number">500</span>)</span><br></pre></td></tr></table></figure><p><img src="/posts/数模国赛2020A题/sub3_2.png"></p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">best_x=[1.06929985e+01 1.97873914e+01 5.68152357e+00 8.90629441e-03 1.99983037e+01]</span><br><span class="line">minArea=446.4150721594619</span><br><span class="line">v=83.71274730906958</span><br><span class="line">tem=[25, 184.78739135418277, 184.78739135418277, 184.78739135418277, 184.78739135418277, 184.78739135418277, 190.68152356719315, 225.00890629441466, 264.99830371980664, 264.99830371980664, 25, 25, 25]</span><br></pre></td></tr></table></figure><h1 id="第四问">第四问</h1><p>把炉温曲线中相同温度的点的时间做平均值，与峰值时间的平方和作为对称性的衡量标准</p><p>然后本问是多目标优化，即阴影面积和对称度</p><p>这里采用的是线性组合作为估值函数，经过多次尝试，发现面积变化始终不大，而对称性比较敏感，最终选择了1:10的比例赋权</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sko.SA <span class="keyword">import</span> SA</span><br><span class="line"><span class="keyword">from</span> sub1 <span class="keyword">import</span> gettT</span><br><span class="line"><span class="keyword">from</span> sub2 <span class="keyword">import</span> checkv</span><br><span class="line"><span class="keyword">from</span> sub3 <span class="keyword">import</span> getarea, X2tem, X2v</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> pandas <span class="keyword">as</span> pd</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">from</span> bisect <span class="keyword">import</span> bisect</span><br><span class="line">exc = pd.read_excel(<span class="string">"input.xlsx"</span>, usecols=range(<span class="number">0</span>, <span class="number">2</span>))</span><br><span class="line">data = exc.values</span><br><span class="line">data = data.T</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">getsy</span><span class="params">(tT)</span>:</span> <span class="comment"># 对称度估值</span></span><br><span class="line"> tT_f = filter(<span class="keyword">lambda</span> x: x[<span class="number">1</span>] >= <span class="number">127</span>, tT)</span><br><span class="line"> tT = np.array(list(tT_f))</span><br><span class="line"> t, T = tT.T.tolist()</span><br><span class="line"> maxT = max(T)</span><br><span class="line"> id = T.index(maxT)</span><br><span class="line"> mt = t[id]</span><br><span class="line"> res = <span class="number">0</span></span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> range(id+<span class="number">1</span>, len(T)):</span><br><span class="line"> j = bisect(T, T[i], <span class="number">0</span>, id)</span><br><span class="line"> res += ((t[i]+t[j])/<span class="number">2</span>-mt)**<span class="number">2</span></span><br><span class="line"> <span class="keyword">return</span> res</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">cost</span><span class="params">(X)</span>:</span></span><br><span class="line"> <span class="keyword">if</span> (X.min() < <span class="number">0</span> <span class="keyword">or</span> X.max() > <span class="number">20</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1e100</span>+np.abs(X.min()<span class="number">-0</span>)+np.abs(X.max()<span class="number">-20</span>)</span><br><span class="line"> v = X2v(X)</span><br><span class="line"> tem = X2tem(X)</span><br><span class="line"> t, T, tT = gettT(v=v, tem=tem, dt=dt)</span><br><span class="line"> <span class="keyword">if</span> (<span class="keyword">not</span> checkv(t, T, tT, dt=dt)):</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1e50</span></span><br><span class="line"> <span class="keyword">return</span> getarea(dt=dt, T=T)+getsy(tT)*<span class="number">1e1</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">'__main__'</span>:</span><br><span class="line"> dt = <span class="number">0.1</span></span><br><span class="line"> sa = SA(func=cost, x0=[<span class="number">0</span>]*<span class="number">5</span>, lb=<span class="number">0</span>, ub=<span class="number">20</span>, max_stay_counter=<span class="number">5000</span>, L=<span class="number">300</span>)</span><br><span class="line"> best_x, best_y = sa.run()</span><br><span class="line"> print(<span class="string">"best_x={}\nbest_y={}"</span>.format(best_x, best_y))</span><br><span class="line"> v = X2v(best_x)</span><br><span class="line"> tem = X2tem(best_x)</span><br><span class="line"> print(<span class="string">"v={}\ntem={}\n"</span>.format(v*<span class="number">60</span>, tem))</span><br><span class="line"> t, T, tT = gettT(v=v, tem=tem, dt=<span class="number">0.1</span>)</span><br><span class="line"> print(<span class="string">"Area="</span>, getarea(dt, T))</span><br><span class="line"> print(<span class="string">"Sy="</span>, getsy(tT))</span><br><span class="line"> plt.plot(t, T)</span><br><span class="line"> plt.plot([<span class="number">0</span>, t.max()], [<span class="number">217</span>, <span class="number">217</span>], <span class="string">'--'</span>)</span><br><span class="line"> plt.savefig(<span class="string">'sub4.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line"> plt.show()</span><br><span class="line"> by_f = filter(<span class="keyword">lambda</span> x: x < <span class="number">1e30</span>, sa.best_y_history)</span><br><span class="line"> by = list(by_f)</span><br><span class="line"> plt.plot(pd.DataFrame(by).cummin(axis=<span class="number">0</span>)) <span class="comment"># 画出最小面积随迭代次数变化的曲线</span></span><br><span class="line"> plt.show()</span><br><span class="line"> plt.savefig(<span class="string">'sub4_2.png'</span>, dpi=<span class="number">500</span>)</span><br></pre></td></tr></table></figure><p><img src="/posts/数模国赛2020A题/sub4.png"></p><p><img src="/posts/数模国赛2020A题/sub4_2.png"></p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">best_x=[12.12066253 2.29087199 0.94809455 19.92827282 19.99803738]</span><br><span class="line">best_y=167663.96859958651</span><br><span class="line">v=86.21115943067932</span><br><span class="line">tem=[25, 167.2908719854776, 167.2908719854776, 167.2908719854776, 167.2908719854776, 167.2908719854776, 185.94809454727962, 244.92827282463267, 264.998037383657, 264.998037383657, 25, 25, 25]</span><br><span class="line">Area= 446.41859958757425</span><br><span class="line">Sy= 16721.754999999896</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<h1 id="第一问">第一问</h1>
<p>忽略电路板厚度，假定温区内部恒温</p>
<p>通过热传导方程得到不同温区之间的空隙中，温度在空间上线性分布</p>
<p>通过牛顿冷却定律和比热容公式解微分方程可知，物体在恒温环境下温度随时间的函数</p>
<p>然后由
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="数模" scheme="http://oi.linkfqy.top/tags/%E6%95%B0%E6%A8%A1/"/>
</entry>
<entry>
<title>SciPy求微分方程数值解</title>
<link href="http://oi.linkfqy.top/posts/SciPy%E6%B1%82%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E6%95%B0%E5%80%BC%E8%A7%A3/"/>
<id>http://oi.linkfqy.top/posts/SciPy求微分方程数值解/</id>
<published>2021-08-14T12:01:12.000Z</published>
<updated>2021-08-14T16:02:07.440Z</updated>
<content type="html"><![CDATA[<h1 id="一阶微分方程组">一阶微分方程（组）</h1><p><code>scipy.integrate.odeint()</code>可以数值解一阶微分方程（组）</p><p>不同于微分方程的符号解，可以给出未知函数的表达式，数值解只能求得指定位置的函数值</p><p><code>sol=odeint(func, y0, t)</code></p><ul><li><code>func</code>：表示一个微分方程<span class="math inline">\(\frac{\text dy}{\text dt} = func(y, t)\)</span>，其中的<code>y</code>可以是向量，以表示微分方程组</li><li><code>y0</code>：未知函数的初值条件，表示<span class="math inline">\(y(t[0])=y_0\)</span>，即第一个自变量取值处的函数值，同样可以是向量</li><li><code>t</code>：自变量的取值序列</li></ul><p>返回值<code>sol</code>是一个<code>len(t)*len(y0)</code>的数组，表示每个取值位置的函数值</p><p>示例： <span class="math display">\[\begin{cases}y'=-2y+x^2+2x \\y(1)=2\end{cases}\]</span> 代码在下面一起给出</p><h1 id="高阶微分方程组">高阶微分方程（组）</h1><p>通过换元可以将高阶微分方程转化为一阶微分方程： <span class="math display">\[y_0=y \\y_1=y_0'=y' \\y_2=y_1'=y'' \\\dots \\y_k=y_{k-1}'=y^{(k)} \\\]</span> 这样就同样可以使用<code>odeint()</code>来获得数值解了</p><p>示例： <span class="math display">\[\begin{cases}y''+2y'+2y=0 \\y(0)=0, y'(0)=1\end{cases}\]</span> 先转化成： <span class="math display">\[\begin{cases}y_0'=y_1 \\y_1'=-2y_1-2y_0 \\y_0(0)=0, y_1(0)=1\end{cases}\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.integrate <span class="keyword">import</span> odeint</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">dy</span><span class="params">(y, x)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> <span class="number">-2</span>*y+x**<span class="number">2</span>+<span class="number">2</span>*x</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">x = np.arange(<span class="number">1</span>, <span class="number">10.5</span>, <span class="number">0.5</span>)</span><br><span class="line">sol = odeint(dy, <span class="number">2</span>, x)</span><br><span class="line">print(sol.T)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">dy</span><span class="params">(y, x)</span>:</span></span><br><span class="line"> y0, y1 = y</span><br><span class="line"> <span class="keyword">return</span> [y1, <span class="number">-2</span>*y1 - <span class="number">2</span>*y0]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">x = np.arange(<span class="number">0</span>, <span class="number">10</span>, <span class="number">0.1</span>)</span><br><span class="line">sol = odeint(dy, [<span class="number">0</span>, <span class="number">1</span>], x)</span><br><span class="line">plt.scatter(x, sol.T[<span class="number">0</span>], marker=<span class="string">'.'</span>) <span class="comment"># 只需要y的0阶导数据</span></span><br><span class="line">plt.savefig(<span class="string">'odeint.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">plt.show()</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">[[ 2. 2.08484933 2.9191691 4.18723381 5.77289452 7.63342241</span><br><span class="line"> 9.75309843 12.12613985 14.75041934 17.62515427 20.75005673 24.12502089</span><br><span class="line"> 27.7500077 31.62500278 35.75000104 40.1250004 44.75000015 49.62500006</span><br><span class="line"> 54.75000002]]</span><br></pre></td></tr></table></figure><p><img src="/posts/SciPy求微分方程数值解/odeint.png"></p><p>Lorenz模型的混沌效应</p><p>Lorenz方程： <span class="math display">\[\begin{cases}\dot x=\sigma(y-x) \\\dot y=\rho x-y-xz \\\dot z=xy-\beta z\end{cases}\]</span> 模拟<span class="math inline">\(\sigma=10,\rho=28,\beta=\frac 8 3\)</span>时<span class="math inline">\(t\in [0,50]\)</span>的运行轨迹，以及给初值微小扰动后解的偏差演化曲线</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.integrate <span class="keyword">import</span> odeint</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">lorenz</span><span class="params">(w, t)</span>:</span></span><br><span class="line"> sigma = <span class="number">10</span></span><br><span class="line"> rho = <span class="number">28</span></span><br><span class="line"> beta = <span class="number">8</span>/<span class="number">3</span></span><br><span class="line"> x, y, z = w</span><br><span class="line"> <span class="keyword">return</span> [sigma*(y-x), rho*x-y-x*z, x*y-beta*z]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">t = np.arange(<span class="number">0</span>, <span class="number">50</span>, <span class="number">0.01</span>)</span><br><span class="line">sol1 = odeint(lorenz, [<span class="number">0.</span>, <span class="number">1.</span>, <span class="number">0.</span>], t)</span><br><span class="line">sol2 = odeint(lorenz, [<span class="number">0.</span>, <span class="number">1.0001</span>, <span class="number">0.</span>], t)</span><br><span class="line"></span><br><span class="line">ax1 = plt.subplot(<span class="number">121</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">ax1.plot(sol1[:, <span class="number">0</span>], sol1[:, <span class="number">1</span>], sol1[:, <span class="number">2</span>], linewidth=<span class="number">1</span>)</span><br><span class="line">ax2 = plt.subplot(<span class="number">122</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">ax2.plot(sol1[:, <span class="number">0</span>]-sol2[:, <span class="number">0</span>], sol1[:, <span class="number">1</span>] -</span><br><span class="line"> sol2[:, <span class="number">1</span>], sol1[:, <span class="number">2</span>]-sol2[:, <span class="number">2</span>], linewidth=<span class="number">1</span>)</span><br><span class="line">plt.savefig(<span class="string">'lorenz.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">plt.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/SciPy求微分方程数值解/lorenz.png"></p>]]></content>
<summary type="html">
<h1 id="一阶微分方程组">一阶微分方程（组）</h1>
<p><code>scipy.integrate.odeint()</code>可以数值解一阶微分方程（组）</p>
<p>不同于微分方程的符号解，可以给出未知函数的表达式，数值解只能求得指定位置的函数值</p>
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>SciPy实现高等数学数值解</title>
<link href="http://oi.linkfqy.top/posts/SciPy%E5%AE%9E%E7%8E%B0%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6%E6%95%B0%E5%80%BC%E8%A7%A3/"/>
<id>http://oi.linkfqy.top/posts/SciPy实现高等数学数值解/</id>
<published>2021-08-13T05:27:40.000Z</published>
<updated>2021-08-13T09:30:38.394Z</updated>
<content type="html"><![CDATA[<h1 id="求导">求导</h1><p><code>scipy.misc.derivative</code>可用于求任意函数的任意阶导数：</p><p><code>def derivative(func, x0, dx=1.0, n=1, order=3)</code></p><ul><li><code>func</code>：自定义的被求导函数<span class="math inline">\(f(x)\)</span></li><li><code>x0</code>：求导的位置，即求<span class="math inline">\(f^{(n)}(x_0)\)</span></li><li><code>dx</code>：用于微分的小量</li><li><code>n</code>：求导的阶数</li><li><code>order</code>：用于求导的采样点个数，必须为>=n+1的奇数</li></ul><p>示例：</p><p>画出<span class="math inline">\(\cos\sqrt{x^2+1}\)</span>及其在<span class="math inline">\(0\)</span>点处的<span class="math inline">\(1,3,5\)</span>阶泰勒展开式 <span class="math display">\[f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i+\dots\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.misc <span class="keyword">import</span> derivative</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(x)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> np.cos(np.sqrt(x**<span class="number">2</span>+<span class="number">1</span>))</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">fac</span><span class="params">(x)</span>:</span></span><br><span class="line"> <span class="keyword">if</span> (x <= <span class="number">1</span>):</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1</span></span><br><span class="line"> <span class="keyword">return</span> fac(x<span class="number">-1</span>)*x</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">dx = <span class="number">1e-4</span></span><br><span class="line">x = np.linspace(<span class="number">-5</span>, <span class="number">5</span>, <span class="number">100</span>)</span><br><span class="line">y = f(x)</span><br><span class="line">plt.plot(x, y, <span class="string">'r-'</span>, linewidth=<span class="number">2</span>)</span><br><span class="line">now = f(np.zeros(<span class="number">100</span>))</span><br><span class="line"><span class="keyword">for</span> i <span class="keyword">in</span> range(<span class="number">1</span>, <span class="number">6</span>):</span><br><span class="line"> now += derivative(f, <span class="number">0</span>, dx, i, order=<span class="number">7</span>)*(x**i)/fac(i)</span><br><span class="line"> <span class="keyword">if</span> (i <span class="keyword">in</span> [<span class="number">1</span>, <span class="number">3</span>, <span class="number">5</span>]):</span><br><span class="line"> plt.plot(x, now, <span class="string">'b--'</span>)</span><br><span class="line">plt.ylim((<span class="number">-2</span>, <span class="number">1</span>))</span><br><span class="line">plt.savefig(<span class="string">"derivative.png"</span>, dpi=<span class="number">500</span>)</span><br><span class="line">plt.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/SciPy实现高等数学数值解/derivative.png"></p><h1 id="定积分">定积分</h1><h2 id="一重积分">一重积分</h2><p><code>scipy.integrate.quad</code>可以求任意函数的一重积分</p><p><code>quad(func, a, b)</code>表示求函数<code>func</code>从<code>a</code>到<code>b</code>的一重积分</p><p>示例：</p><p>求<span class="math inline">\(\int_0^\infty e^{-x}\sin(x^2+2)\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.integrate <span class="keyword">import</span> quad</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(x)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> np.exp(-x)*np.sin(x**<span class="number">2</span>+<span class="number">2</span>)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">print(quad(f, <span class="number">0</span>, np.inf))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">(0.37378979243874044, 1.3876530327162045e-08)</span><br></pre></td></tr></table></figure><p>返回的值分别是积分结果和绝对误差</p><h2 id="多重积分">多重积分</h2><p><code>scipy.integrate.dblquad</code>和<code>scipy.integrate.tplquad</code>分别能够求二重和三重积分</p><p><code>dblquad(func,a,b,gfun,hfun)</code>表示<span class="math inline">\(\int_a^b \text dx \int_{gfun(x)}^{hfun(x)} func(x,y) \text dy\)</span></p><p><code>tplquad(func,a,b,gfun,hfun,qfun,rfun)</code>表示<span class="math inline">\(\int_a^b \text dx \int_{gfun(x)}^{hfun(x)} \text dy \int_{qfun(x,y)}^{rfun(x,y)} func(x,y,z) \text dz\)</span></p><p>返回值和<code>quad</code>一致</p><p>需要注意的是，<code>func</code>定义的参数顺序必须是<code>func(x,y,z)</code></p><h1 id="非线性方程组">非线性方程（组）</h1><p><code>scipy.optimize.fsolve</code>可以解非线性方程（组）</p><p><code>fsolve(func,x0)</code>表示求<span class="math inline">\(func(x)=0\)</span>的根，迭代初值<code>x0</code></p><p>若要解方程组，可将<code>func(x)</code>的<code>x</code>定义为向量</p><p>示例： <span class="math display">\[x^3+1.1x^2+0.9x-1.4=0 \\\begin{cases}5x_2+3=0 \\4x_1^2-2\sin(x_2x_3)=0 \\x_2x_3-1.5=0\end{cases}\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> fsolve</span><br><span class="line"></span><br><span class="line">print(fsolve(<span class="keyword">lambda</span> x: x**<span class="number">3</span>+<span class="number">1.1</span>*x**<span class="number">2</span>+<span class="number">0.9</span>*x<span class="number">-1.4</span>, <span class="number">0</span>))</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(x)</span>:</span></span><br><span class="line"> x1, x2, x3 = x.tolist()</span><br><span class="line"> <span class="keyword">return</span> [<span class="number">5</span>*x2+<span class="number">3</span>, <span class="number">4</span>*x1**<span class="number">2</span> - <span class="number">2</span>*np.sin(x2*x3), x2*x3 - <span class="number">1.5</span>]</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">print(fsolve(f, [<span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>]))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">[0.67065731]</span><br><span class="line">[-0.70622057 -0.6 -2.5 ]</span><br></pre></td></tr></table></figure><h1 id="求极值点">求极值点</h1><p><code>scipy.optimize.minimize</code>是求非线性规划的函数，若不加约束，即可求任意函数（一元，多元）的极小值点</p><p><code>minimize(func,x0)</code>表示求<span class="math inline">\(func(x)=0\)</span>的最小值，迭代初值<code>x0</code></p><p>返回的是一个<code>result</code>类，可以调用成员以获取相应信息</p><p>示例： <span class="math display">\[f_1(x)=e^x\cos(2x) \\f_2(x_1,x_2)=100(x_2-x_1^2)^2+(1-x_1)^2\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> minimize</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f1</span><span class="params">(x)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> np.exp(x)*np.cos(<span class="number">2</span>*x)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f2</span><span class="params">(x)</span>:</span></span><br><span class="line"> x1, x2 = x.tolist()</span><br><span class="line"> <span class="keyword">return</span> <span class="number">100</span>*(x2-x1**<span class="number">2</span>)**<span class="number">2</span>+(<span class="number">1</span>-x1)**<span class="number">2</span></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">print(minimize(f1, <span class="number">0</span>), <span class="string">'\n'</span>)</span><br><span class="line">print(minimize(f2, [<span class="number">2</span>, <span class="number">2</span>]))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><span class="line"> fun: -0.2344426467071059</span><br><span class="line">hess_inv: array([[0.85441485]])</span><br><span class="line"> jac: array([-1.86264515e-08])</span><br><span class="line"> message: 'Optimization terminated successfully.'</span><br><span class="line"> nfev: 14</span><br><span class="line"> nit: 5</span><br><span class="line"> njev: 7</span><br><span class="line"> status: 0</span><br><span class="line"> success: True</span><br><span class="line"> x: array([-1.33897255]) </span><br><span class="line"></span><br><span class="line"> fun: 1.8932893809017893e-11</span><br><span class="line">hess_inv: array([[0.51675994, 1.03186494],</span><br><span class="line"> [1.03186494, 2.0655726 ]])</span><br><span class="line"> jac: array([ 5.27380711e-06, -2.50575298e-06])</span><br><span class="line"> message: 'Optimization terminated successfully.'</span><br><span class="line"> nfev: 105</span><br><span class="line"> nit: 30</span><br><span class="line"> njev: 35</span><br><span class="line"> status: 0</span><br><span class="line"> success: True</span><br><span class="line"> x: array([0.99999565, 0.99999129])</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<h1 id="求导">求导</h1>
<p><code>scipy.misc.derivative</code>可用于求任意函数的任意阶导数：</p>
<p><code>def derivative(func, x0, dx=1.0, n=1, order=3)</code
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>NumPy解线性代数问题</title>
<link href="http://oi.linkfqy.top/posts/NumPy%E8%A7%A3%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E9%97%AE%E9%A2%98/"/>
<id>http://oi.linkfqy.top/posts/NumPy解线性代数问题/</id>
<published>2021-08-11T15:51:32.000Z</published>
<updated>2021-08-12T15:05:31.680Z</updated>
<content type="html"><![CDATA[<h1 id="矩阵运算">矩阵运算</h1><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy.linalg <span class="keyword">as</span> la</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">a = np.array([[<span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>]])</span><br><span class="line">b = np.array([[<span class="number">4</span>], [<span class="number">5</span>], [<span class="number">6</span>]])</span><br><span class="line">print(la.norm(a)) <span class="comment"># a的二范数，即模长</span></span><br><span class="line">print(b@a) <span class="comment"># 即a.dot(b)，可表示点乘（一维数组）和矩阵乘法（二维数组）</span></span><br><span class="line">a = np.reshape(a, <span class="number">-1</span>)</span><br><span class="line">b = np.reshape(b, <span class="number">-1</span>)</span><br><span class="line">print(np.inner(a, b)) <span class="comment"># a,b的内积，两参数的shape必须为(n,)（1维）</span></span><br><span class="line">print(np.cross(a, b)) <span class="comment"># a,b的叉乘</span></span><br><span class="line"></span><br><span class="line">A = np.arange(<span class="number">1</span>, <span class="number">17</span>).reshape(<span class="number">4</span>, <span class="number">4</span>)</span><br><span class="line">B = np.eye(<span class="number">4</span>) <span class="comment"># 4阶单位矩阵</span></span><br><span class="line">print(la.det(A)) <span class="comment"># 行列式</span></span><br><span class="line">print(la.matrix_rank(A)) <span class="comment"># 秩</span></span><br><span class="line">print(A.T) <span class="comment"># 转置阵</span></span><br><span class="line">print(la.inv(A+<span class="number">10</span>*B)) <span class="comment"># (A+10*B)的逆矩阵</span></span><br><span class="line">print(A@B)</span><br><span class="line">print(np.c_[A, B]) <span class="comment"># 横向分块矩阵</span></span><br><span class="line">print(np.r_[A, B]) <span class="comment"># 纵向分块矩阵</span></span><br><span class="line">print(A[<span class="number">0</span>:<span class="number">2</span>, <span class="number">0</span>:<span class="number">2</span>]) <span class="comment"># 矩阵切片，左上角2*2的子矩阵</span></span><br><span class="line">print(np.delete(A, <span class="number">3</span>, axis=<span class="number">0</span>)) <span class="comment"># 删除某（些）行(0)或列(1)</span></span><br><span class="line">print(np.delete(A, [<span class="number">0</span>, <span class="number">2</span>], axis=<span class="number">1</span>))</span><br><span class="line">print(np.delete(A, [<span class="number">0</span>, <span class="number">2</span>])) <span class="comment"># 默认按一维顺序删除单点</span></span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><span class="line">3.7416573867739413</span><br><span class="line">[[ 4 8 12]</span><br><span class="line"> [ 5 10 15]</span><br><span class="line"> [ 6 12 18]]</span><br><span class="line">32</span><br><span class="line">[-3 6 -3]</span><br><span class="line">4.7331654313261276e-30</span><br><span class="line">2</span><br><span class="line">[[ 1 5 9 13]</span><br><span class="line"> [ 2 6 10 14]</span><br><span class="line"> [ 3 7 11 15]</span><br><span class="line"> [ 4 8 12 16]]</span><br><span class="line">[[ 0.11277778 0.00333333 -0.00611111 -0.01555556]</span><br><span class="line"> [-0.005 0.09 -0.015 -0.02 ]</span><br><span class="line"> [-0.02277778 -0.02333333 0.07611111 -0.02444444]</span><br><span class="line"> [-0.04055556 -0.03666667 -0.03277778 0.07111111]]</span><br><span class="line">[[ 1. 2. 3. 4.]</span><br><span class="line"> [ 5. 6. 7. 8.]</span><br><span class="line"> [ 9. 10. 11. 12.]</span><br><span class="line"> [13. 14. 15. 16.]]</span><br><span class="line">[[ 1. 2. 3. 4. 1. 0. 0. 0.]</span><br><span class="line"> [ 5. 6. 7. 8. 0. 1. 0. 0.]</span><br><span class="line"> [ 9. 10. 11. 12. 0. 0. 1. 0.]</span><br><span class="line"> [13. 14. 15. 16. 0. 0. 0. 1.]]</span><br><span class="line">[[ 1. 2. 3. 4.]</span><br><span class="line"> [ 5. 6. 7. 8.]</span><br><span class="line"> [ 9. 10. 11. 12.]</span><br><span class="line"> [13. 14. 15. 16.]</span><br><span class="line"> [ 1. 0. 0. 0.]</span><br><span class="line"> [ 0. 1. 0. 0.]</span><br><span class="line"> [ 0. 0. 1. 0.]</span><br><span class="line"> [ 0. 0. 0. 1.]]</span><br><span class="line">[[1 2]</span><br><span class="line"> [5 6]]</span><br><span class="line">[[ 1 2 3 4]</span><br><span class="line"> [ 5 6 7 8]</span><br><span class="line"> [ 9 10 11 12]]</span><br><span class="line">[[ 2 4]</span><br><span class="line"> [ 6 8]</span><br><span class="line"> [10 12]</span><br><span class="line"> [14 16]]</span><br><span class="line">[ 2 4 5 6 7 8 9 10 11 12 13 14 15 16]</span><br></pre></td></tr></table></figure><h1 id="解线性方程组">解线性方程组</h1><h2 id="齐次线性方程组的基础解系">齐次线性方程组的基础解系</h2><p><code>scipy.linalg.null_space()</code>可以获得基础解系 <span class="math display">\[\begin{cases}x_1 -5x_2 +2x_3 -3x_4=0 \\5x_1 +3x_2 +6x_3 -x_4=0 \\2_1 +4x_2 +2x_3 +x_4=0 \\\end{cases}\]</span> 的基础解系：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.linalg <span class="keyword">import</span> null_space</span><br><span class="line">A = np.array([[<span class="number">1</span>, <span class="number">-5</span>, <span class="number">2</span>, <span class="number">-3</span>], [<span class="number">5</span>, <span class="number">3</span>, <span class="number">6</span>, <span class="number">-1</span>], [<span class="number">2</span>, <span class="number">4</span>, <span class="number">2</span>, <span class="number">1</span>]])</span><br><span class="line">print(null_space(A)) <span class="comment"># 零空间，即基础解系</span></span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">[[ 0.79428706 -0.07832183]</span><br><span class="line"> [-0.19204896 -0.36561063]</span><br><span class="line"> [-0.52695834 0.38844091]</span><br><span class="line"> [ 0.23353839 0.84220438]]</span><br></pre></td></tr></table></figure><p>两个列向量即为基础解系</p><h2 id="非齐次线性方程组解">非齐次线性方程组解</h2><p><code>numpy.linalg.pinv()</code>可以求矩阵的伪逆矩阵，这样对于形如<span class="math inline">\(Ax=b\)</span>的线性方程组，<code>pinv(A)@b</code>就可以得到一组解</p><p>一般地，当方程组有：</p><ul><li>无穷多组解，得到的是最小范数解</li><li>唯一解，得到的是唯一的那组解</li><li>无解，得到的是最小二乘解<span class="math inline">\(x^*\)</span>，即满足<span class="math inline">\(|Ax^*-b|^2\)</span>最小的解</li></ul><h1 id="特征值与特征向量">特征值与特征向量</h1><p><code>numpy.linalg.eig()</code>可以得到矩阵的特征值和特征向量</p><p>返回二元组分别是特征值和特征向量的list，特征向量表示为列向量</p><p>例如： $$ A=,</p><p>P=,</p><p>=$$ 验证<span class="math inline">\(P^{-1}AP=\Lambda\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> numpy.linalg <span class="keyword">import</span> eig</span><br><span class="line">A = np.array([[<span class="number">0</span>, <span class="number">-2</span>, <span class="number">2</span>], [<span class="number">-2</span>, <span class="number">-3</span>, <span class="number">4</span>], [<span class="number">2</span>, <span class="number">4</span>, <span class="number">-3</span>]])</span><br><span class="line">values, vectors = eig(A)</span><br><span class="line">print(values)</span><br><span class="line">print(vectors)</span><br><span class="line">print(np.linalg.inv(vectors)@A@vectors)</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">[ 1. -8. 1.]</span><br><span class="line">[[ 0.94280904 -0.33333333 -0.2981424 ]</span><br><span class="line"> [-0.23570226 -0.66666667 0.74535599]</span><br><span class="line"> [ 0.23570226 0.66666667 0.59628479]]</span><br><span class="line">[[ 1.00000000e+00 -6.10622664e-16 -1.11022302e-16]</span><br><span class="line"> [ 0.00000000e+00 -8.00000000e+00 -4.44089210e-16]</span><br><span class="line"> [-5.55111512e-17 4.44089210e-16 1.00000000e+00]]</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<h1 id="矩阵运算">矩阵运算</h1>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="lin
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="NumPy" scheme="http://oi.linkfqy.top/tags/NumPy/"/>
</entry>
<entry>
<title>Python实现拟合</title>
<link href="http://oi.linkfqy.top/posts/Python%E5%AE%9E%E7%8E%B0%E6%8B%9F%E5%90%88/"/>
<id>http://oi.linkfqy.top/posts/Python实现拟合/</id>
<published>2021-08-07T13:30:55.000Z</published>
<updated>2021-08-07T15:22:15.173Z</updated>
<content type="html"><![CDATA[<p>拟合：给定<span class="math inline">\(n\)</span>个数据点，求一<span class="math inline">\(f(x)\)</span>使其与数据点最“接近”，即： <span class="math display">\[J=\sum_{i=1}^n(fx_i-y_i)^2\]</span> 最小，称为最小二乘拟合。</p><p>往往需要提前指定<span class="math inline">\(f(x)\)</span>的类型，常用的有某次多项式、双曲函数、指数函数等</p><p><code>scipy.optimize.curve_fit</code>可以实现任意类型函数的拟合</p><p>用法：</p><p><code>curve_fit(f, xdata, ydata, p0=None)</code></p><ul><li><code>f</code>为一函数：<code>f(x,...)</code>，<code>x</code>为自变量，后面的参数全为拟合函数的待定参数</li><li><code>xdata</code>和<code>ydata</code>是数据点的坐标向量</li><li><code>p0</code>为待定参数的初值向量</li><li>D维情形，x是D维向量，<code>xdata</code>是D维向量构成的序列，p0同</li></ul><p>返回二元组<code>(popt,pcov)</code>：</p><ul><li><code>popt</code>：一维数组，表示拟合函数的参数</li><li><code>pcov</code>：二维阵列，<code>popt</code>的估计协方差。对角线提供参数估计的方差。</li></ul><p>示例：</p><p>用二次多项式拟合一元函数</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> curve_fit</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(x, a, b, c)</span>:</span> <span class="keyword">return</span> a*x**<span class="number">2</span>+b*x+c</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">x = np.arange(<span class="number">0</span>, <span class="number">1.1</span>, <span class="number">0.1</span>)</span><br><span class="line">y = np.array([<span class="number">-0.447</span>, <span class="number">1.978</span>, <span class="number">3.28</span>, <span class="number">6.16</span>, <span class="number">7.08</span>,</span><br><span class="line"> <span class="number">7.34</span>, <span class="number">7.66</span>, <span class="number">9.56</span>, <span class="number">9.48</span>, <span class="number">9.30</span>, <span class="number">11.2</span>])</span><br><span class="line">popt, pcov = curve_fit(f, x, y)</span><br><span class="line">print(<span class="string">"a={}, b={}, c={}"</span>.format(*popt)) <span class="comment"># 打印参数</span></span><br><span class="line">print(<span class="string">"f(0.25)={},f(0.35)={}"</span>.format(</span><br><span class="line"> *f(np.array([<span class="number">0.25</span>, <span class="number">0.35</span>]), *popt))) <span class="comment"># 打印预测值</span></span><br><span class="line">plt.scatter(x, y)</span><br><span class="line">xn = np.linspace(<span class="number">0</span>, <span class="number">1</span>, <span class="number">100</span>)</span><br><span class="line">yn = f(xn, *popt)</span><br><span class="line">plt.plot(xn, yn)</span><br><span class="line">plt.savefig(<span class="string">'1.png'</span>, dpi=<span class="number">500</span>), plt.show()</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">a=-9.810839009366013, b=20.129292913034863, c=-0.03167107877459929</span><br><span class="line">f(0.25)=4.387474711398741,f(0.35)=5.811753662140266</span><br></pre></td></tr></table></figure><p><img src="/posts/Python实现拟合/1.png"></p><p>对曲面<span class="math inline">\(z=e^{-\frac{(x-\mu_1)^2+(y-\mu_2)^2}{2\sigma^2}}\)</span>添加噪声，再进行拟合，其中<span class="math inline">\(\mu_1=1,\mu_2=2,\sigma=3\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> curve_fit</span><br><span class="line"><span class="keyword">import</span> pylab <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">f</span><span class="params">(x, u1, u2, sgm)</span>:</span></span><br><span class="line"> <span class="keyword">return</span> np.exp(-((x[<span class="number">0</span>]-u1)**<span class="number">2</span>+(x[<span class="number">1</span>]-u2)**<span class="number">2</span>)/(<span class="number">2</span>*sgm**<span class="number">2</span>))</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">x = np.linspace(<span class="number">-6</span>, <span class="number">6</span>, <span class="number">200</span>)</span><br><span class="line">y = np.linspace(<span class="number">-8</span>, <span class="number">8</span>, <span class="number">300</span>)</span><br><span class="line">X, Y = np.meshgrid(x, y) <span class="comment"># 获得网格坐标矩阵</span></span><br><span class="line">X = np.reshape(X, (<span class="number">1</span>, <span class="number">-1</span>))</span><br><span class="line">Y = np.reshape(Y, (<span class="number">1</span>, <span class="number">-1</span>)) <span class="comment"># 降维</span></span><br><span class="line">xy = np.vstack((X, Y)) <span class="comment"># 转换成坐标点序列的形式</span></span><br><span class="line">z = f(xy, <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>) <span class="comment"># 求原始函数的各点值</span></span><br><span class="line">zr = z+<span class="number">0.2</span>*np.random.normal(size=z.shape) <span class="comment"># 生成噪声数据</span></span><br><span class="line">popt, pcov = curve_fit(f, xy, zr)</span><br><span class="line">print(<span class="string">"u1={}, u2={}, sgm={}"</span>.format(*popt)) <span class="comment"># 打印参数</span></span><br><span class="line">zn = f(xy, *popt) <span class="comment"># 求拟合函数的各点值</span></span><br><span class="line">X = np.reshape(X, (<span class="number">200</span>, <span class="number">300</span>))</span><br><span class="line">Y = np.reshape(Y, (<span class="number">200</span>, <span class="number">300</span>))</span><br><span class="line">zn = np.reshape(zn, (<span class="number">200</span>, <span class="number">300</span>))</span><br><span class="line">ax = plt.axes(projection=<span class="string">'3d'</span>)</span><br><span class="line">ax.plot_surface(X, Y, zn, cmap=<span class="string">'gist_rainbow'</span>)</span><br><span class="line">plt.savefig(<span class="string">'2.png'</span>, dpi=<span class="number">500</span>), plt.show()</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">u1=1.0055785483402968, u2=2.002582704331591, sgm=3.0134682873777496</span><br></pre></td></tr></table></figure><p><img src="/posts/Python实现拟合/2.png"></p>]]></content>
<summary type="html">
<p>拟合：给定<span class="math inline">\(n\)</span>个数据点，求一<span class="math inline">\(f(x)\)</span>使其与数据点最“接近”，即： <span class="math display">\[
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>Python实现插值</title>
<link href="http://oi.linkfqy.top/posts/Python%E5%AE%9E%E7%8E%B0%E6%8F%92%E5%80%BC/"/>
<id>http://oi.linkfqy.top/posts/Python实现插值/</id>
<published>2021-08-06T16:04:37.000Z</published>
<updated>2021-08-07T07:12:05.611Z</updated>
<content type="html"><![CDATA[<p>插值：求一函数<span class="math inline">\(f(x)\)</span>使其穿过给定的所有数据点<span class="math inline">\((x_i,y_i)\)</span></p><p>在数据分析中很有用</p><h1 id="拉格朗日插值lagrange">拉格朗日插值（Lagrange）</h1><p>拉格朗日插值的<span class="math inline">\(f(x)\)</span>为<span class="math inline">\(n-1\)</span>次多项式，<span class="math inline">\(n\)</span>为数据点个数</p><p>可以证明，任意<span class="math inline">\(n\)</span>个数据点的<span class="math inline">\(n-1\)</span>次插值函数存在且唯一</p><p>具体来说， <span class="math display">\[f(x)=\sum_{i=0}^n l_i(x)y_i \\l_i(x)=\prod_{j=0,j\neq i}^n \frac {x-x_j} {x_i-x_j}\]</span> 由此公式容易用Python实现拉格朗日插值</p><p>但是一次性将所有数据点进行插值，次数过高，不仅复杂度高，误差也大，因此常进行分段插值，即用分段函数表示<span class="math inline">\(f(x)\)</span></p><h1 id="样条插值spline">样条插值（spline）</h1><p>单纯的分段插值得到的函数性质往往不够好（光滑性）</p><p>对于划分：<span class="math inline">\(\Delta:a=x_0<x_1<\dots<x_n=b\)</span>，<span class="math inline">\(m\)</span>阶样条插值函数需要满足：</p><ul><li>在每个小区间<span class="math inline">\([x_i,x_{i+1}](i=0,1,\dots,n-1)\)</span>上是<span class="math inline">\(m\)</span>次多项式</li><li>在区间<span class="math inline">\([a,b]\)</span>上具有<span class="math inline">\(m-1\)</span>阶导数</li></ul><p>显然阶梯插值为0阶样条插值，折线插值为1阶样条插值</p><p>一般直接使用<code>scipy.interpolate</code>实现的样条插值</p><h1 id="scipy.interpolate求解插值问题"><code>scipy.interpolate</code>求解插值问题</h1><h2 id="一维插值interp1d">一维插值（<code>interp1d</code>）</h2><p><code>interp1d(x,y,kind='linear')</code></p><p>前两个参数为数据点向量，kind指定插值方法：</p><table><thead><tr class="header"><th style="text-align: center;">kind</th><th style="text-align: center;">说明</th></tr></thead><tbody><tr class="odd"><td style="text-align: center;"><code>'next','previous'</code></td><td style="text-align: center;">阶梯插值，值为最近的前/后一个<span class="math inline">\(y_i\)</span></td></tr><tr class="even"><td style="text-align: center;"><code>'zero', 'linear', 'quadratic', 'cubic'</code></td><td style="text-align: center;">0，1，2，3阶样条插值</td></tr><tr class="odd"><td style="text-align: center;"><code>0,1,2,3,5,...</code></td><td style="text-align: center;">用<code>int</code>指定样条插值的阶数，3以上只能为奇数</td></tr></tbody></table><p>返回一个插值函数，可以传入<span class="math inline">\(x\)</span>返回<span class="math inline">\(f(x)\)</span>，也可以传入<span class="math inline">\(\vec x\)</span>返回<span class="math inline">\(f(\vec x)\)</span></p><p>示例：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">import</span> matplotlib.pyplot <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">from</span> scipy.interpolate <span class="keyword">import</span> interp1d</span><br><span class="line">x = np.arange(<span class="number">0</span>, <span class="number">25</span>, <span class="number">2</span>)</span><br><span class="line">y = np.array([<span class="number">12</span>, <span class="number">9</span>, <span class="number">9</span>, <span class="number">10</span>, <span class="number">18</span>, <span class="number">24</span>, <span class="number">28</span>, <span class="number">27</span>, <span class="number">25</span>, <span class="number">20</span>, <span class="number">18</span>, <span class="number">15</span>, <span class="number">13</span>])</span><br><span class="line">xnew = np.linspace(<span class="number">0</span>, <span class="number">24</span>, <span class="number">500</span>)</span><br><span class="line">f0 = interp1d(x, y, <span class="number">0</span>) <span class="comment"># 'zero'</span></span><br><span class="line">y0 = f0(xnew)</span><br><span class="line">f1 = interp1d(x, y)</span><br><span class="line">y1 = f1(xnew)</span><br><span class="line">f3 = interp1d(x, y, <span class="string">'cubic'</span>)</span><br><span class="line">y3 = f3(xnew)</span><br><span class="line">f5 = interp1d(x, y, <span class="number">5</span>)</span><br><span class="line">y5 = f5(xnew)</span><br><span class="line">plt.rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">plt.rc(<span class="string">'font'</span>, family=<span class="string">'SimHei'</span>)</span><br><span class="line">plt.subplot(<span class="number">221</span>), plt.plot(xnew, y0), plt.xlabel(<span class="string">"（A）分段阶梯插值"</span>)</span><br><span class="line">plt.subplot(<span class="number">222</span>), plt.plot(xnew, y1), plt.xlabel(<span class="string">"（B）分段线性插值"</span>)</span><br><span class="line">plt.subplot(<span class="number">223</span>), plt.plot(xnew, y3), plt.xlabel(<span class="string">"（C）三次样条插值"</span>)</span><br><span class="line">plt.subplot(<span class="number">224</span>), plt.plot(xnew, y5), plt.xlabel(<span class="string">"（D）五次样条插值"</span>)</span><br><span class="line">plt.savefig(<span class="string">"interp1d.png"</span>, dpi=<span class="number">500</span>), plt.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python实现插值/interp1d.png"></p><h2 id="二维插值interp2d">二维插值（<code>interp2d</code>）</h2><p><code>interp2d(x,y,z,kind='linear')</code>接受网格分布的数据点进行插值</p><p>x，y为坐标向量，z为数据矩阵</p><p>返回一个插值函数，输入x，y向量即可获得数据矩阵</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> IPython.core.pylabtools <span class="keyword">import</span> figsize</span><br><span class="line"><span class="keyword">import</span> matplotlib.pyplot <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.interpolate <span class="keyword">import</span> interp2d</span><br><span class="line"><span class="keyword">from</span> mpl_toolkits <span class="keyword">import</span> mplot3d</span><br><span class="line">z = np.loadtxt(<span class="string">"data.txt"</span>)</span><br><span class="line">x = np.arange(<span class="number">0</span>, <span class="number">1500</span>, <span class="number">100</span>)</span><br><span class="line">y = np.arange(<span class="number">1200</span>, <span class="number">-100</span>, <span class="number">-100</span>) <span class="comment"># 文件里的y是倒序输入的</span></span><br><span class="line">f = interp2d(x, y, z, kind=<span class="string">'cubic'</span>)</span><br><span class="line">xn = np.linspace(<span class="number">0</span>, <span class="number">1400</span>, <span class="number">141</span>)</span><br><span class="line">yn = np.linspace(<span class="number">0</span>, <span class="number">1200</span>, <span class="number">121</span>)</span><br><span class="line">zn = f(xn, yn)</span><br><span class="line">plt.rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">plt.subplot(<span class="number">1</span>, <span class="number">2</span>, <span class="number">1</span>)</span><br><span class="line">plt.contour(xn, yn, zn)</span><br><span class="line">plt.xlabel(<span class="string">'$x$'</span>), plt.ylabel(<span class="string">'$y$'</span>, rotation=<span class="number">90</span>)</span><br><span class="line">ax = plt.subplot(<span class="number">1</span>, <span class="number">2</span>, <span class="number">2</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">X, Y = np.meshgrid(xn, yn)</span><br><span class="line">ax.plot_surface(X, Y, zn, cmap=<span class="string">'viridis'</span>)</span><br><span class="line">ax.set_xlabel(<span class="string">'$x$'</span>), ax.set_ylabel(<span class="string">'$y$'</span>), ax.set_zlabel(<span class="string">'$z$'</span>)</span><br><span class="line">plt.savefig(<span class="string">'interp2d.png'</span>, dpi=<span class="number">500</span>), plt.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python实现插值/interp2d.png"></p><h2 id="散点插值griddata">散点插值（<code>griddata</code>）</h2><p>若数据点并无分布规律，可用<code>griddata</code>进行插值</p><p>可处理任意D维的数据点</p><p><code>griddata(points, values, xi, method='linear')</code></p><p><code>points</code>为数据点位置（D维向量）的序列，<code>values</code>为数据点值的序列</p><p><code>xi</code>为需要插值的空间，形式与<code>points</code>相同，或用格点坐标矩阵的D元组表示</p><p><code>method</code>插值方法：<code>'nearest','linear','cubic'</code>分别对应0，1，3阶插值</p><p>返回<code>xi</code>中的数据值</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> mpl_toolkits <span class="keyword">import</span> mplot3d</span><br><span class="line"><span class="keyword">import</span> matplotlib.pyplot <span class="keyword">as</span> plt</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> scipy.interpolate <span class="keyword">import</span> griddata</span><br><span class="line">x = np.array([<span class="number">129</span>, <span class="number">140</span>, <span class="number">103.5</span>, <span class="number">88</span>, <span class="number">185.5</span>, <span class="number">195</span>, <span class="number">105</span>,</span><br><span class="line"> <span class="number">157.5</span>, <span class="number">107.5</span>, <span class="number">77</span>, <span class="number">81</span>, <span class="number">162</span>, <span class="number">162</span>, <span class="number">117.5</span>])</span><br><span class="line">y = np.array([<span class="number">7.5</span>, <span class="number">141.5</span>, <span class="number">23</span>, <span class="number">147</span>, <span class="number">22.5</span>, <span class="number">137.5</span>, <span class="number">85.5</span>, -</span><br><span class="line"> <span class="number">6.5</span>, <span class="number">-81</span>, <span class="number">3</span>, <span class="number">56.5</span>, <span class="number">-66.5</span>, <span class="number">84</span>, <span class="number">-33.5</span>])</span><br><span class="line">z = -np.array([<span class="number">4</span>, <span class="number">8</span>, <span class="number">6</span>, <span class="number">8</span>, <span class="number">6</span>, <span class="number">8</span>, <span class="number">8</span>, <span class="number">9</span>, <span class="number">9</span>, <span class="number">8</span>, <span class="number">8</span>, <span class="number">9</span>, <span class="number">4</span>, <span class="number">9</span>])</span><br><span class="line">xy = np.array([x, y]).T</span><br><span class="line">xn = np.linspace(x.min(), x.max(), <span class="number">100</span>)</span><br><span class="line">yn = np.linspace(y.min(), y.max(), <span class="number">100</span>)</span><br><span class="line">X, Y = np.meshgrid(xn, yn)</span><br><span class="line">zn = griddata(xy, z, (X, Y), method=<span class="string">'nearest'</span>)</span><br><span class="line">ax = plt.subplot(<span class="number">121</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">ax.plot_surface(X, Y, zn, cmap=<span class="string">'viridis'</span>)</span><br><span class="line">ax.set_xlabel(<span class="string">'$x$'</span>), ax.set_ylabel(<span class="string">'$y$'</span>), ax.set_zlabel(<span class="string">'$z$'</span>)</span><br><span class="line">plt.subplot(<span class="number">122</span>)</span><br><span class="line">a = plt.contourf(X, Y, zn, <span class="number">8</span>, cmap=plt.cm.Spectral)</span><br><span class="line">plt.clabel(plt.contour(xn, yn, zn))</span><br><span class="line">plt.colorbar(a)</span><br><span class="line">plt.savefig(<span class="string">'griddata.png'</span>, dpi=<span class="number">500</span>), plt.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python实现插值/griddata.png"></p>]]></content>
<summary type="html">
<p>插值：求一函数<span class="math inline">\(f(x)\)</span>使其穿过给定的所有数据点<span class="math inline">\((x_i,y_i)\)</span></p>
<p>在数据分析中很有用</p>
<h1 id=
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>SciPy解非线性规划</title>
<link href="http://oi.linkfqy.top/posts/SciPy%E8%A7%A3%E9%9D%9E%E7%BA%BF%E6%80%A7%E8%A7%84%E5%88%92/"/>
<id>http://oi.linkfqy.top/posts/SciPy解非线性规划/</id>
<published>2021-08-05T15:59:25.000Z</published>
<updated>2021-08-05T16:51:59.725Z</updated>
<content type="html"><![CDATA[<p>与线性规划不同，非线性规划的目标函数和约束都不一定是线性的</p><p>因此最优解也不一定在可行域边界上，一般而言用迭代算法得到的也只是局部最优解</p><p>标准模型如下： <span class="math display">\[\min f(\vec x) \\\text{s.t.}\begin{cases} g_i(\vec x) \geq 0,\ \ \ i=1,2,...,m, \\ h_i(\vec x) = 0,\ \ \ i=1,2,...,l,\end{cases}\]</span> <code>scipy.optimize.minimize</code>可以解决非线性规划问题</p><p>用法：</p><p><code>minimize(fun,x0,bounds=None,constraints=None)</code></p><p>其中<code>fun</code>为目标函数，输入一个向量<span class="math inline">\(\vec x\)</span>返回对应值</p><p><code>x0</code>为迭代初值，决定是否能取到全局最优解</p><p><code>bounds</code>与线性规划的<code>linprog</code>一样</p><p><code>constraints</code>为约束条件，具体实现为单个dict或dict的序列，每个dict应有两个键值：</p><ul><li><p><code>type</code>：字符串<code>'eq'</code>和<code>'ineq'</code>之一，表示该约束为等号约束或不等号约束</p></li><li><p><code>fun</code>：函数，传入向量<span class="math inline">\(\vec x\)</span>，返回约束式的左值，在可行解中该值大于等于0（不等号约束）/等于0（等号约束）</p><p>特别地，若有一系列约束为线性约束，则可将一系列右值视为一个向量，利用<code>numpy.array</code>重载的比较符号，</p><p>将这些约束缩为一个，即<code>lambda x:b-A@x</code></p></li></ul>]]></content>
<summary type="html">
<p>与线性规划不同，非线性规划的目标函数和约束都不一定是线性的</p>
<p>因此最优解也不一定在可行域边界上，一般而言用迭代算法得到的也只是局部最优解</p>
<p>标准模型如下： <span class="math display">\[
\min f(\vec x)
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>SciPy解决线性规划问题</title>
<link href="http://oi.linkfqy.top/posts/SciPy%E8%A7%A3%E5%86%B3%E7%BA%BF%E6%80%A7%E8%A7%84%E5%88%92%E9%97%AE%E9%A2%98/"/>
<id>http://oi.linkfqy.top/posts/SciPy解决线性规划问题/</id>
<published>2021-08-04T12:25:14.000Z</published>
<updated>2021-08-04T15:55:17.828Z</updated>
<content type="html"><![CDATA[<p><code>scipy.optimize.linprog</code>可以解决线性规划问题</p><p>标准模型： <span class="math display">\[\min_x \ z= c^T x, \\ \mbox{s.t.} \ \begin{cases} A x \leq b,\\ A_{eq} x = b_{eq},\\ l \leq x \leq u . \end{cases}\]</span> 其中<span class="math inline">\(A\)</span>和<span class="math inline">\(A_{eq}\)</span>分别是不等约束和等号约束的系数矩阵，<span class="math inline">\(l\)</span>和<span class="math inline">\(u\)</span>是上下界向量</p><p>用法：</p><p><code>linprog(c, A=None, b=None, Aeq=None, beq=None, bounds=None)</code></p><p>前5个参数对应标准模型，bounds是一个由<code>(li,ui)</code>组成的序列，表示每个<code>xi</code>的边界</p><p>当<code>bounds</code>为<code>None</code>时，默认下界全为0，上界全为正无穷</p><p>返回的对象记为<code>res</code>，具有以下属性：</p><p><code>res.fun</code>为目标函数的最小值，<code>res.x</code>为最优解</p><p>示例： <span class="math display">\[\min_x \ z= -x_1+4x_2, \\ \mbox{s.t.} \ \begin{cases} -3x_1+x_2 \leq 6,\\ x_1+2x_2 \leq 4,\\ x_2 \geq -3 . \end{cases}\]</span></p><p>两种方式求解：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> scipy.optimize <span class="keyword">import</span> linprog</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">c = [<span class="number">-1</span>, <span class="number">4</span>]</span><br><span class="line">A = [[<span class="number">-3</span>, <span class="number">1</span>], [<span class="number">1</span>, <span class="number">2</span>]]</span><br><span class="line">b = [<span class="number">6</span>, <span class="number">4</span>]</span><br><span class="line">bounds = [[<span class="keyword">None</span>, <span class="keyword">None</span>], [<span class="number">-3</span>, <span class="keyword">None</span>]]</span><br><span class="line"><span class="comment"># A = [[-3, 1], [1, 2], [0, -1]]</span></span><br><span class="line"><span class="comment"># b = [6, 4, 3]</span></span><br><span class="line"><span class="comment"># bounds = [[None, None], [None, None]] # 与bounds=None区别</span></span><br><span class="line">res = linprog(c, A, b, <span class="keyword">None</span>, <span class="keyword">None</span>, bounds)</span><br><span class="line">print(res.fun)</span><br><span class="line">print(res.x)</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">-21.99999984082492</span><br><span class="line">[ 9.99999989 -2.99999999]</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><code>scipy.optimize.linprog</code>可以解决线性规划问题</p>
<p>标准模型： <span class="math display">\[
\min_x \ z= c^T x, \\
\mbox{s.t.} \
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SciPy" scheme="http://oi.linkfqy.top/tags/SciPy/"/>
</entry>
<entry>
<title>SymPy实现高等数学符号解</title>
<link href="http://oi.linkfqy.top/posts/SymPy%E5%AE%9E%E7%8E%B0%E9%AB%98%E7%AD%89%E6%95%B0%E5%AD%A6%E7%AC%A6%E5%8F%B7%E8%A7%A3/"/>
<id>http://oi.linkfqy.top/posts/SymPy实现高等数学符号解/</id>
<published>2021-08-03T01:42:19.000Z</published>
<updated>2021-08-04T01:44:00.847Z</updated>
<content type="html"><![CDATA[<p>一些前置知识：</p><p><code>x=symbols('x')</code>声明变量<span class="math inline">\(x\)</span></p><p><code>y.subs(x,x0)</code><span class="math inline">\(y\)</span>是<span class="math inline">\(x\)</span>的表达式时，带入<span class="math inline">\(x=x_0\)</span>的值</p><p><code>y.n()</code>常量表达式求浮点值</p><h1 id="求极限">求极限</h1><p><code>limit(e, z, z0, dir="+")</code></p><p><span class="math inline">\(e\)</span>是<span class="math inline">\(z\)</span>的表达式，求<span class="math inline">\(e(z)\)</span>在<span class="math inline">\(z_0\)</span>处的极限，<span class="math inline">\(\text{dir}\)</span>可以指定是<span class="math inline">\(z_0^+\)</span>还是<span class="math inline">\(z_0^-\)</span></p><p>以下计算了<span class="math inline">\(\lim_{x\rightarrow 0}\frac{\sin x}x\)</span>，<span class="math inline">\(\lim_{x\rightarrow \infty} (1+\frac 1 x)^x\)</span>，<span class="math inline">\(\lim_{x\rightarrow 0}\sin(\frac 1 x)\)</span>三个极限</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">x = symbols(<span class="string">'x'</span>)</span><br><span class="line">print(limit(sin(x)/x, x, <span class="number">0</span>))</span><br><span class="line">print(limit(pow(<span class="number">1</span>+<span class="number">1</span>/x, x), x, oo))</span><br><span class="line">print(limit(sin(<span class="number">1</span>/x), x, <span class="number">0</span>))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">1</span><br><span class="line">E</span><br><span class="line">AccumBounds(-1, 1)</span><br></pre></td></tr></table></figure><h1 id="求导数">求导数</h1><p><code>diff(expr,x1,n1,x2,n2,...)</code></p><p>表达式<code>expr</code>依次对<code>xi</code>求<code>ni</code>阶偏导，<code>ni</code>不写默认为1</p><p><span class="math inline">\(z=\sin x+x^2e^y\)</span>，求<span class="math inline">\(\frac{\partial^2 z}{\partial x^2},\frac{\partial z}{\partial y},\frac{\partial^2 z}{\partial x\partial y}\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">x, y = symbols(<span class="string">"x y"</span>)</span><br><span class="line">z = sin(x)+x**<span class="number">2</span>*exp(y)</span><br><span class="line">print(diff(z, x, <span class="number">2</span>))</span><br><span class="line">print(diff(z, y))</span><br><span class="line">print(diff(z, x, y))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">2*exp(y) - sin(x)</span><br><span class="line">x**2*exp(y)</span><br><span class="line">2*x*exp(y)</span><br></pre></td></tr></table></figure><h1 id="级数求和">级数求和</h1><p><code>summation(expr,(i,a,b))</code></p><p>expr是关于i的函数，求<span class="math inline">\(\sum_{i=a}^b expr(i)\)</span></p><p>示例：<span class="math inline">\(\sum_{k=1}^n k^2\)</span>和<span class="math inline">\(\sum_{k=1}^\infty \frac 1 {k^2}\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">k, n = symbols(<span class="string">'k n'</span>)</span><br><span class="line">print(factor(summation(k**<span class="number">2</span>, (k, <span class="number">1</span>, n)))) <span class="comment"># 计算结果因式分解</span></span><br><span class="line">print(summation(<span class="number">1</span>/k**<span class="number">2</span>, (k, <span class="number">1</span>, oo)))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">n*(n + 1)*(2*n + 1)/6</span><br><span class="line">pi**2/6</span><br></pre></td></tr></table></figure><h1 id="泰勒展开">泰勒展开</h1><p><code>series(expr, x=None, x0=0, n=6, dir="+")</code></p><p><span class="math inline">\(expr\)</span>是<span class="math inline">\(x\)</span>的表达式，在<span class="math inline">\(x_0\)</span>处的<span class="math inline">\(n\)</span>阶展开，<span class="math inline">\(\text{dir}\)</span>可以指定是<span class="math inline">\(x_0^+\)</span>还是<span class="math inline">\(x_0^-\)</span></p><p>下面演示了<span class="math inline">\(\sin x\)</span>在0点的3，5，7阶泰勒展开式</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> pylab <span class="keyword">import</span> rc</span><br><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">from</span> sympy.plotting <span class="keyword">import</span> *</span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">x = symbols(<span class="string">'x'</span>)</span><br><span class="line">y = sin(x)</span><br><span class="line">yi = list(range(<span class="number">8</span>))</span><br><span class="line"><span class="keyword">for</span> k <span class="keyword">in</span> range(<span class="number">3</span>, <span class="number">8</span>, <span class="number">2</span>):</span><br><span class="line"> print(series(y, x, <span class="number">0</span>, k))</span><br><span class="line"> yi[k] = series(y, x, <span class="number">0</span>, k).removeO() <span class="comment"># 带有余项的表达式无法作图</span></span><br><span class="line">plot(y, yi[<span class="number">3</span>], yi[<span class="number">5</span>], yi[<span class="number">7</span>], (x, <span class="number">0</span>, <span class="number">2</span>),</span><br><span class="line"> xlabel=<span class="string">'$x$'</span>, ylabel=<span class="string">'$y$'</span>).save(<span class="string">'series.png'</span>)</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">x + O(x**3)</span><br><span class="line">x - x**3/6 + O(x**5)</span><br><span class="line">x - x**3/6 + x**5/120 + O(x**7)</span><br></pre></td></tr></table></figure><p><img src="/posts/SymPy实现高等数学符号解/series.png"></p><h1 id="积分">积分</h1><p>不定积分：</p><p><code>integrate(f,x)</code></p><p>相当于<span class="math inline">\(\int f(x)\mathrm d x\)</span></p><p>定积分：</p><p><code>integrate(f,(x,a,b))</code></p><p>相当于<span class="math inline">\(\int_a^b f(x)\mathrm d x\)</span></p><p>以下代码求解<span class="math inline">\(\int_0^\pi \sin(2x)\mathrm d x,\int_0^{+\infty} \frac {\sin(x)} x\mathrm d x,\int \sin(x)e^x\mathrm d x\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">x = symbols(<span class="string">'x'</span>)</span><br><span class="line">print(integrate(sin(<span class="number">2</span>*x), (x, <span class="number">0</span>, pi)))</span><br><span class="line">print(integrate(sin(x)/x, (x, <span class="number">0</span>, oo)))</span><br><span class="line">print(integrate(sin(x)*exp(x), x))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">0</span><br><span class="line">pi/2</span><br><span class="line">exp(x)*sin(x)/2 - exp(x)*cos(x)/2</span><br></pre></td></tr></table></figure><h1 id="解代数方程组">解代数方程（组）</h1><p><code>solve(expr,x)</code></p><p>求解方程<span class="math inline">\(expr=0\)</span>，其中<span class="math inline">\(x\)</span>是自变量</p><p>返回一个list存放所有的根</p><p><code>solve([...,expr_i,...],[...,x_i,...])</code></p><p>求解方程组，其中<span class="math inline">\(expr_i\)</span>对应<span class="math inline">\(x_i\)</span></p><p>返回一个list存放所有根，每个根用一个tuple表示，对应此根所有各个变量的值</p><p><code>roots(expr,x)</code></p><p>返回一个dict，表示每个根以及它的重数</p><p>示例： <span class="math display">\[x^3-1=0 \\(x-2)^2(x-1)^3=0 \\\begin{equation} \begin{cases} x^2+y^2=1 \\ x-y=0 \end{cases}\end{equation}\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">x, y = symbols(<span class="string">"x y"</span>)</span><br><span class="line">print(solve(x**<span class="number">3</span><span class="number">-1</span>, x))</span><br><span class="line">print(solve((x<span class="number">-2</span>)**<span class="number">2</span>*(x<span class="number">-1</span>)**<span class="number">3</span>, x))</span><br><span class="line">print(roots((x<span class="number">-2</span>)**<span class="number">2</span>*(x<span class="number">-1</span>)**<span class="number">3</span>, x))</span><br><span class="line">print(solve([x**<span class="number">2</span>+y**<span class="number">2</span><span class="number">-1</span>, x-y], [x, y]))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]</span><br><span class="line">[1, 2]</span><br><span class="line">{2: 2, 1: 3}</span><br><span class="line">[(-sqrt(2)/2, -sqrt(2)/2), (sqrt(2)/2, sqrt(2)/2)]</span><br></pre></td></tr></table></figure><h1 id="微分方程组的通解">微分方程（组）的通解</h1><p>微分方程中的<span class="math inline">\(y\)</span>是关于<span class="math inline">\(x\)</span>的函数，而且是未知函数，而非未知量，因此要将<span class="math inline">\(y\)</span>声明为Function</p><p><code>y=Function('y')</code></p><p><code>y=symbols('y',cls=Function)</code></p><p>两种写法都是可以的</p><p><code>dsolve(eq,f(x),ics=)</code>，其中<code>eq</code>是关于<code>f(x)</code>的表达式，求解<span class="math inline">\(eq=0\)</span></p><p>ics可以以一个dict指定f(x)的初值</p><p>示例： <span class="math display">\[y''-5y'+6y=0 \\y''-5y'+6y=xe^{2x} \\y''-5y'+6y=0,y(0)=1,y'(0)=0 \\y''-5y'+6y=xe^{2x},y(0)=1,y(2)=0\]</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> *</span><br><span class="line">x = symbols(<span class="string">'x'</span>)</span><br><span class="line">y = Function(<span class="string">'y'</span>)</span><br><span class="line">eq1 = diff(y(x), x, <span class="number">2</span>)<span class="number">-5</span>*diff(y(x), x)+<span class="number">6</span>*y(x)</span><br><span class="line">eq2 = diff(y(x), x, <span class="number">2</span>)<span class="number">-5</span>*diff(y(x), x)+<span class="number">6</span>*y(x)-x*exp(<span class="number">2</span>*x)</span><br><span class="line">print(dsolve(eq1, y(x)))</span><br><span class="line">print(dsolve(eq2, y(x)))</span><br><span class="line">print(dsolve(eq1, y(x), ics={y(<span class="number">0</span>): <span class="number">1</span>, diff(y(x), x).subs(x, <span class="number">0</span>): <span class="number">0</span>}))</span><br><span class="line">print(dsolve(eq2, y(x), ics={y(<span class="number">0</span>): <span class="number">1</span>, y(<span class="number">2</span>): <span class="number">0</span>}))</span><br></pre></td></tr></table></figure><p>输出：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">Eq(y(x), (C1 + C2*exp(x))*exp(2*x))</span><br><span class="line">Eq(y(x), (C1 + C2*exp(x) - x**2/2 - x)*exp(2*x))</span><br><span class="line">Eq(y(x), (3 - 2*exp(x))*exp(2*x))</span><br><span class="line">Eq(y(x), (-x**2/2 - x - 3*exp(x)/(1 - exp(2)) + (4 - exp(2))/(1 - exp(2)))*exp(2*x))</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p>一些前置知识：</p>
<p><code>x=symbols('x')</code>声明变量<span class="math inline">\(x\)</span></p>
<p><code>y.subs(x,x0)</code><span class="math
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="SymPy" scheme="http://oi.linkfqy.top/tags/SymPy/"/>
</entry>
<entry>
<title>SymPy符号制图</title>
<link href="http://oi.linkfqy.top/posts/SymPy%E7%AC%A6%E5%8F%B7%E5%88%B6%E5%9B%BE/"/>
<id>http://oi.linkfqy.top/posts/SymPy符号制图/</id>
<published>2021-08-02T07:03:59.000Z</published>
<updated>2021-08-03T01:43:38.730Z</updated>
<content type="html"><![CDATA[<p>与Matplotlib不同，SymPy无需离散的数据，仅需符号表达式即可制图</p><h1 id="二维曲线">二维曲线</h1><p>单个图像：</p><p><code>plot(expr, range, **kwargs)</code></p><p>其中<code>expr</code>为表达式，表达式由<code>sympy.abc</code>导入的符号组成</p><p><code>range</code>为三元组<code>(x,l,r)</code>，指定了自变量以及上下界</p><p><code>**kwargs</code>指定的参数与<code>matplotlib</code>类似，这里不做过多介绍</p><p>多个图像在同一范围：</p><p><code>plot(expr1, expr2, ..., range, **kwargs)</code></p><p>多个图像在不同范围：</p><p><code>plot((expr1, range1), (expr2, range2), ..., **kwargs)</code></p><p>以下代码画出了<span class="math inline">\(y_1=2\sin x,\ x\in [-6,6]\)</span>和<span class="math inline">\(y_2=\cos(x+\frac \pi 4),\ x\in [-5,5]\)</span>的图像</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sympy.plotting <span class="keyword">import</span> plot</span><br><span class="line"><span class="keyword">from</span> sympy.abc <span class="keyword">import</span> x, pi</span><br><span class="line"><span class="keyword">from</span> sympy.functions <span class="keyword">import</span> sin, cos</span><br><span class="line">p = plot((<span class="number">2</span>*sin(x), (x, <span class="number">-6</span>, <span class="number">6</span>)), (cos(x+pi/<span class="number">4</span>), (x, <span class="number">-5</span>, <span class="number">5</span>)),</span><br><span class="line"> show=<span class="keyword">False</span>, xlabel=<span class="string">'$x$'</span>, ylabel=<span class="string">'$y$'</span>)</span><br><span class="line">p.save(<span class="string">"plot.png"</span>)</span><br><span class="line">p.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/SymPy符号制图/plot.png"></p><h1 id="三维曲线">三维曲线</h1><p><span class="math inline">\(z=\sin(\sqrt{x^2+y^2})\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> pylab <span class="keyword">import</span> rc</span><br><span class="line"><span class="keyword">from</span> sympy.plotting <span class="keyword">import</span> plot3d</span><br><span class="line"><span class="keyword">from</span> sympy.abc <span class="keyword">import</span> x, y</span><br><span class="line"><span class="keyword">from</span> sympy.functions <span class="keyword">import</span> sin, sqrt</span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">p = plot3d(sin(sqrt(x**<span class="number">2</span>+y**<span class="number">2</span>)), show=<span class="keyword">False</span>, xlabel=<span class="string">'$x$'</span>, ylabel=<span class="string">'$y$'</span>)</span><br><span class="line">p.save(<span class="string">'plot3d.png'</span>)</span><br><span class="line">p.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/SymPy符号制图/plot3d.png"></p><h1 id="隐函数画图">隐函数画图</h1><p>用<code>sumpy.Eq</code>表示一个方程</p><p><code>Eq(expr_a,expr_b)</code>表示等式两边分别是<code>expr_a,expr_b</code></p><p><code>Eq(expr)</code>等价于<code>Eq(expr,0)</code></p><p><span class="math inline">\((x-1)^2+(y-2)^2-4=0\)</span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> pylab <span class="keyword">import</span> rc</span><br><span class="line"><span class="keyword">from</span> sympy <span class="keyword">import</span> plot_implicit, Eq</span><br><span class="line"><span class="keyword">from</span> sympy.abc <span class="keyword">import</span> x, y</span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">p = plot_implicit(Eq((x<span class="number">-1</span>)**<span class="number">2</span>+(y<span class="number">-2</span>)**<span class="number">2</span><span class="number">-4</span>), show=<span class="keyword">False</span>,</span><br><span class="line"> xlabel=<span class="string">'$x$'</span>, ylabel=<span class="string">'$y$'</span>)</span><br><span class="line">p.save(<span class="string">'plot_implicit.png'</span>)</span><br><span class="line">p.show()</span><br></pre></td></tr></table></figure><p><img src="/posts/SymPy符号制图/plot_implicit.png"></p>]]></content>
<summary type="html">
<p>与Matplotlib不同，SymPy无需离散的数据，仅需符号表达式即可制图</p>
<h1 id="二维曲线">二维曲线</h1>
<p>单个图像：</p>
<p><code>plot(expr, range, **kwargs)</code></p>
<p>其中
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="Sympy" scheme="http://oi.linkfqy.top/tags/Sympy/"/>
</entry>
<entry>
<title>Python使用Matplotlib作图</title>
<link href="http://oi.linkfqy.top/posts/Python%E4%BD%BF%E7%94%A8Matplotlib%E4%BD%9C%E5%9B%BE/"/>
<id>http://oi.linkfqy.top/posts/Python使用Matplotlib作图/</id>
<published>2021-08-01T12:43:30.000Z</published>
<updated>2021-08-03T01:43:55.801Z</updated>
<content type="html"><![CDATA[<p><code>Matplotlib.pyplot</code>将原本面向对象的作图操作封装成函数，使之更近似于MATLAB的交互式命令</p><p>只要有离散的数据，就可以做出柱状、散点、饼状、折线、3D图形等可视化图表</p><h1 id="柱状图bar">柱状图（bar）</h1><p><code>pyplot.bar()</code>只需提供两个数组作为各数据的横纵坐标，即可生成柱状图</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib <span class="keyword">import</span> rcParams</span><br><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">y = np.array((<span class="number">2100</span>, <span class="number">3000</span>, <span class="number">4000</span>))</span><br><span class="line">x = np.array([<span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>])</span><br><span class="line">width = <span class="number">0.2</span></span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">bar(x, y, width) <span class="comment"># 指定柱宽</span></span><br><span class="line">ylabel(<span class="string">"消费数据"</span>)</span><br><span class="line">xticks(x, [<span class="string">"用户A"</span>, <span class="string">"用户B"</span>, <span class="string">"用户C"</span>], rotation=<span class="number">20</span>) <span class="comment"># 将x标签改为用户名，并旋转20度显示</span></span><br><span class="line">rcParams[<span class="string">'font.sans-serif'</span>] = [<span class="string">'SimHei'</span>] <span class="comment"># 设置中文字体，否则会乱码</span></span><br><span class="line">savefig(<span class="string">'bar.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show() <span class="comment"># 显示在屏幕</span></span><br></pre></td></tr></table></figure><p>效果图：</p><p><img src="/posts/Python使用Matplotlib作图/bar.png"></p><p>若要显示负号，需<code>rcParams['axes.unicode_minus']=False</code></p><h1 id="散点图scatter">散点图（scatter）</h1><p>与<code>bar</code>类似</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> numpy.linalg.linalg <span class="keyword">import</span> _eigvalsh_dispatcher</span><br><span class="line"><span class="keyword">from</span> pylab <span class="keyword">import</span> * <span class="comment"># pylab=pyplot+部分numpy函数</span></span><br><span class="line">x = array(range(<span class="number">1</span>, <span class="number">8</span>))</span><br><span class="line">y = <span class="string">"27.0 26.5 26.3 26.1 25.7 25.3 24.8"</span></span><br><span class="line">y = <span class="string">","</span>.join(y.split())</span><br><span class="line">y = array(eval(y))</span><br><span class="line">scatter(x, y, marker=<span class="string">'*'</span>) <span class="comment"># 用星形显示散点</span></span><br><span class="line">savefig(<span class="string">"scatter.png"</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/scatter.png"></p><h1 id="折线图plot">折线图（plot）</h1><p>基本用法为<code>plot(x,y,s)</code>，前两个参数为数据点的横纵坐标，s为指定线条颜色、线条样式和数据点形状的字符串</p><p>也可在同一个坐标内显示多个折线</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line">x = np.linspace(<span class="number">0</span>, <span class="number">2</span>*np.pi, <span class="number">200</span>)</span><br><span class="line">y1 = np.sin(x)</span><br><span class="line">y2 = np.cos(pow(x, <span class="number">2</span>))</span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line"><span class="comment"># rc('text', usetex=True) # 可以用LaTeX字体显示</span></span><br><span class="line">plot(x, y1, <span class="string">'r-'</span>, label=<span class="string">'$sin(x)$'</span>, linewidth=<span class="number">2</span>) <span class="comment"># 红色实线</span></span><br><span class="line">plot(x, y2, <span class="string">'b--'</span>, label=<span class="string">'$cos(x^2)$'</span>) <span class="comment"># 蓝色虚线</span></span><br><span class="line">xlabel(<span class="string">'$x$'</span>)</span><br><span class="line">ylabel(<span class="string">'$y$'</span>, rotation=<span class="number">0</span>) <span class="comment"># y标签默认旋转90度</span></span><br><span class="line">rc(<span class="string">'legend'</span>, loc=<span class="string">'lower left'</span>) <span class="comment"># 图例在左下显示</span></span><br><span class="line">legend()</span><br><span class="line">savefig(<span class="string">'plot.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/plot.png"></p><h1 id="饼状图pie">饼状图（pie）</h1><p>只需一维数组</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib <span class="keyword">import</span> rcParams</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line">a = np.array([<span class="number">124</span>, <span class="number">45</span>, <span class="number">72</span>, <span class="number">87</span>, <span class="number">11</span>])</span><br><span class="line">pie(a, labels=[<span class="string">"A产品"</span>, <span class="string">"B产品"</span>, <span class="string">"C产品"</span>, <span class="string">"D产品"</span>, <span class="string">"E产品"</span>])</span><br><span class="line">title(<span class="string">"销售额分析"</span>)</span><br><span class="line">rcParams[<span class="string">'font.sans-serif'</span>] = [<span class="string">'SimHei'</span>]</span><br><span class="line">savefig(<span class="string">"pie.png"</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/pie.png"></p><h1 id="子窗口subplot">子窗口（subplot）</h1><p><code>subplot(nrows, ncols, index)</code></p><p>前两个参数将整个窗口分为<code>nrows*ncols</code>的区域</p><p>函数返回从上到下，从左到右的第<code>index</code>个区域（从1开始）</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line">x = np.linspace(<span class="number">0</span>, <span class="number">2</span>*np.pi, <span class="number">200</span>)</span><br><span class="line">y1 = np.sin(x)</span><br><span class="line">y2 = np.cos(x)</span><br><span class="line">y3 = np.sin(x*x)</span><br><span class="line">rc(<span class="string">'font'</span>, size=<span class="number">16</span>)</span><br><span class="line">ax1 = subplot(<span class="number">2</span>, <span class="number">2</span>, <span class="number">1</span>) <span class="comment"># 创建左上窗口</span></span><br><span class="line">ax1.plot(x, y1, <span class="string">'r'</span>, label=<span class="string">'$sin(x)$'</span>)</span><br><span class="line">legend()</span><br><span class="line"></span><br><span class="line">ax2 = subplot(<span class="number">2</span>, <span class="number">2</span>, <span class="number">2</span>) <span class="comment"># 创建右上窗口</span></span><br><span class="line">ax2.plot(x, y2, <span class="string">'b--'</span>, label=<span class="string">'$cos(x)$'</span>)</span><br><span class="line">legend()</span><br><span class="line"></span><br><span class="line">ax3 = subplot(<span class="number">2</span>, <span class="number">1</span>, <span class="number">2</span>) <span class="comment"># 创建下方窗口</span></span><br><span class="line">ax3.plot(x, y3, <span class="string">'k--'</span>, label=<span class="string">'$sin(x^2)$'</span>)</span><br><span class="line">legend()</span><br><span class="line">savefig(<span class="string">'subplot.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/subplot.png"></p><p>上方的两个窗口分别占据了2*2划分的前两个区域</p><p>下方窗口占据了2*1划分的第二个区域</p><h1 id="三维曲线plot3d">三维曲线（plot3D）</h1><p>以下代码画出<span class="math inline">\(x=t\sin t,y=t\cos t,z=t\)</span>的图形</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">ax = axes(projection=<span class="string">'3d'</span>) <span class="comment"># 先设置三维模式</span></span><br><span class="line">z = np.linspace(<span class="number">0</span>, <span class="number">100</span>, <span class="number">1000</span>)</span><br><span class="line">x = np.sin(z)*z</span><br><span class="line">y = np.cos(z)*z</span><br><span class="line">ax.plot3D(x, y, z, <span class="string">'k'</span>)</span><br><span class="line">savefig(<span class="string">'plot3D.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/plot3D.png"></p><h1 id="三维曲面plot_surfaceplot_wireframe">三维曲面（plot_surface,plot_wireframe）</h1><p><code>plot_surface</code>生成表面图形，<code>plot_wireframe</code>生成网格图形</p><p>生成三维曲面时，需指定网格坐标（描点坐标）</p><p>以下代码画出<span class="math inline">\(z=\sin (\sqrt{x^2+y^2})\)</span>的图形</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">x = np.linspace(<span class="number">-6</span>, <span class="number">6</span>, <span class="number">30</span>)</span><br><span class="line">y = np.linspace(<span class="number">-6</span>, <span class="number">6</span>, <span class="number">30</span>)</span><br><span class="line">X, Y = np.meshgrid(x, y) <span class="comment"># 生成网格坐标矩阵</span></span><br><span class="line">Z = np.sin(np.sqrt(X**<span class="number">2</span>+Y**<span class="number">2</span>))</span><br><span class="line"></span><br><span class="line">ax1 = subplot(<span class="number">1</span>, <span class="number">2</span>, <span class="number">1</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">ax1.plot_surface(X, Y, Z, cmap=<span class="string">'viridis'</span>) <span class="comment"># 使用viridis的配色创建表面图形</span></span><br><span class="line">ax1.set_xlabel(<span class="string">'x'</span>)</span><br><span class="line">ax1.set_ylabel(<span class="string">'y'</span>)</span><br><span class="line">ax1.set_zlabel(<span class="string">'z'</span>)</span><br><span class="line"></span><br><span class="line">ax2 = subplot(<span class="number">1</span>, <span class="number">2</span>, <span class="number">2</span>, projection=<span class="string">'3d'</span>)</span><br><span class="line">ax2.plot_wireframe(X, Y, Z, color=<span class="string">'c'</span>)</span><br><span class="line">ax2.set_xlabel(<span class="string">'x'</span>)</span><br><span class="line">ax2.set_ylabel(<span class="string">'y'</span>)</span><br><span class="line">ax2.set_zlabel(<span class="string">'z'</span>)</span><br><span class="line"></span><br><span class="line">savefig(<span class="string">'plot_surface.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/plot_surface.png"></p><h1 id="等高线图contour">等高线图（contour）</h1><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">z = np.loadtxt(<span class="string">"data.txt"</span>) <span class="comment"># 从文件导入高程数据</span></span><br><span class="line">x = np.arange(<span class="number">0</span>, <span class="number">1500</span>, <span class="number">100</span>)</span><br><span class="line">y = np.arange(<span class="number">1200</span>, <span class="number">-10</span>, <span class="number">-100</span>)</span><br><span class="line">contr = contour(x, y, z)</span><br><span class="line">clabel(contr) <span class="comment"># 标注高度</span></span><br><span class="line">savefig(<span class="string">'contour.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/contour.png"></p><h1 id="向量图quiver">向量图（quiver）</h1><p>以下代码画出向量场<span class="math inline">\(\vec A=(y\cos x,y\sin x)\)</span></p><p><code>quiver(x, y, vx, vy)</code>分别接收两个网格坐标矩阵，以及向量场的xy分量</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> matplotlib.pyplot <span class="keyword">import</span> *</span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">x = np.linspace(<span class="number">0</span>, <span class="number">15</span>, <span class="number">11</span>)</span><br><span class="line">y = np.linspace(<span class="number">0</span>, <span class="number">10</span>, <span class="number">12</span>)</span><br><span class="line">x, y = np.meshgrid(x, y)</span><br><span class="line">vx = y*np.cos(x)</span><br><span class="line">vy = y*np.sin(x)</span><br><span class="line">quiver(x, y, vx, vy)</span><br><span class="line">savefig(<span class="string">'quiver.png'</span>, dpi=<span class="number">500</span>)</span><br><span class="line">show()</span><br></pre></td></tr></table></figure><p><img src="/posts/Python使用Matplotlib作图/quiver.png"></p>]]></content>
<summary type="html">
<p><code>Matplotlib.pyplot</code>将原本面向对象的作图操作封装成函数，使之更近似于MATLAB的交互式命令</p>
<p>只要有离散的数据，就可以做出柱状、散点、饼状、折线、3D图形等可视化图表</p>
<h1 id="柱状图bar">柱状图（
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
<category term="Matplotlib" scheme="http://oi.linkfqy.top/tags/Matplotlib/"/>
</entry>
<entry>
<title>LitePal使用指南</title>
<link href="http://oi.linkfqy.top/posts/LitePal%E4%BD%BF%E7%94%A8%E6%8C%87%E5%8D%97/"/>
<id>http://oi.linkfqy.top/posts/LitePal使用指南/</id>
<published>2021-05-03T09:19:42.000Z</published>
<updated>2021-05-03T09:19:47.991Z</updated>
<content type="html"><![CDATA[<h1 id="litepal使用指南">LitePal使用指南</h1><p>我们项目中只涉及轻量级的数据库操作，因此使用LitePal中封装好的接口可以大大简化操作</p><p><strong>建议全程使用LitePal进行开发</strong></p><p><a href="https://github.com/guolindev/LitePal" target="_blank" rel="noopener">github地址</a></p><h2 id="配置环境">配置环境</h2><p><strong>注意：以下有些步骤是我已经完成的，请务必先将本地工程更新至仓库中的最新版本</strong></p><p>在<code>app/build.gradle</code>中添加对LitePal的引用：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">dependencies {</span><br><span class="line"> implementation 'org.litepal.guolindev:core:3.2.3'</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>注意使用最新版本3.2.3</p><p>在<code>assets</code>中创建<code>litepal.xml</code>：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><?xml version="1.0" encoding="utf-8"?></span><br><span class="line"><litepal></span><br><span class="line"> <dbname value="main" /></span><br><span class="line"> <version value="1" /></span><br><span class="line"> <list></span><br><span class="line"> <mapping class="com.hitsz.eazytime.model.Todo" /></span><br><span class="line"> </list></span><br><span class="line"></litepal></span><br></pre></td></tr></table></figure><p>其中<code>dbname</code>指定数据库名，<code>version</code>指定数据库版本（增减数据表、增减数据表字段、改变字段类型等操作都必须增加版本号）</p><p>在<code>list</code>中使用<code>mapping</code>标签关联每个数据表</p><p>在<code>AndroidManifest.xml</code>配置：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><manifest></span><br><span class="line"> <application</span><br><span class="line"> android:name="org.litepal.LitePalApplication"</span><br><span class="line"> ...</span><br><span class="line"> ></span><br><span class="line"> ...</span><br><span class="line"> </application></span><br><span class="line"></manifest></span><br></pre></td></tr></table></figure><h2 id="创建数据表">创建数据表</h2><p>LitePal将数据表和java类一一对应，因此对对象的操作就很容易转化成对数据表的操作</p><p>在<code>com.hitsz.eazytime.model</code>包下新增一个java文件用于定义类：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><span class="line">public class Todo extends LitePalSupport {</span><br><span class="line"> private String title;</span><br><span class="line"> private Date startTime;</span><br><span class="line"> private Date endTime;</span><br><span class="line"> int id;</span><br><span class="line"> </span><br><span class="line"> public Todo(){</span><br><span class="line"> //构造函数</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public String getTitle() {</span><br><span class="line"> return title;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public void setTitle(String title) {</span><br><span class="line"> this.title = title;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public Date getStartTime() {</span><br><span class="line"> return startTime;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public void setStartTime(Date startTime) {</span><br><span class="line"> this.startTime = startTime;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public Date getEndTime() {</span><br><span class="line"> return endTime;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public void setEndTime(Date endTime) {</span><br><span class="line"> this.endTime = endTime;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> public int getId() {</span><br><span class="line"> return id;</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure><p>LitePal支持的数据类型一共有8种：<code>int、short、long、float、double、boolean、String、Date</code></p><p><code>id</code>字段：无论类中定义与否，数据表里都会存在，且作为自增主键使用，但在类中定义<code>id</code>及<code>getId</code>可以方便查询</p><p><strong>代码规范：类中的所有成员必须以private定义，取值和修改必须经过get和set函数</strong></p><p><strong>注意：新版本LitePal的数据表类应当继承自<code>LitePalSupport</code>而非<code>DataSupport</code></strong></p><p><strong>注意：修改了类的定义后，请务必增加数据库的版本号，否则对类的更改无法应用至数据库</strong></p><h2 id="增查删改">增查删改</h2><p>这部分在《第一行代码》和<a href="https://blog.csdn.net/guolin_blog/category_2522725.html" target="_blank" rel="noopener">作者博客</a>里有较详细的介绍，这里有重点地讲一下</p><h3 id="增">增</h3><p>一般new一个新的对象出来，然后save就可以了</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">Todo todo=new Todo(...);</span><br><span class="line">todo.save();</span><br></pre></td></tr></table></figure><h3 id="查">查</h3><p><strong>注意：新版的查询使用<code>LitePal</code>而非<code>DataSupport</code></strong></p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">List<Todo> todos = LitePal.select(...).where(...).order(...).limit(...).offset(...).find(Todo.class)</span><br></pre></td></tr></table></figure><p>以上连缀无固定顺序，也可不写其中的某些连缀</p><p>注：LitePal本身支持关联表的激进查询，但不推荐使用，合理的写法是在类中实现一个返回关联字段的函数，详细请看作者博客</p><h3 id="删">删</h3><p>没什么要说的，见作者博客</p><h3 id="改">改</h3><p>没什么要说的，见作者博客</p>]]></content>
<summary type="html">
<h1 id="litepal使用指南">LitePal使用指南</h1>
<p>我们项目中只涉及轻量级的数据库操作，因此使用LitePal中封装好的接口可以大大简化操作</p>
<p><strong>建议全程使用LitePal进行开发</strong></p>
<p><a
</summary>
<category term="Android" scheme="http://oi.linkfqy.top/categories/Android/"/>
<category term="数据库" scheme="http://oi.linkfqy.top/tags/%E6%95%B0%E6%8D%AE%E5%BA%93/"/>
</entry>
<entry>
<title>简易版git教程</title>
<link href="http://oi.linkfqy.top/posts/%E7%AE%80%E6%98%93%E7%89%88Git%E6%95%99%E7%A8%8B/"/>
<id>http://oi.linkfqy.top/posts/简易版Git教程/</id>
<published>2021-03-30T14:07:25.000Z</published>
<updated>2021-03-30T14:25:24.253Z</updated>
<content type="html"><![CDATA[<h1 id="简易版git教程">简易版git教程</h1><p>这是写给大一立项小组成员看的简易版git教程</p><p>我们做这个项目只需要一些基本的git命令就可以了，因此这里教大家10min速成git</p><h2 id="准备工作">准备工作</h2><ul><li>GitHub和git的关系</li></ul><p>git是版本管理工具，适合多人合作，储存代码的地方叫做仓库，仓库可以在本地，也可以在学校的服务器上，GitHub是个托管仓库的网站，大家经常把代码放在GitHub上</p><p>在<a href="https://git-scm.com/downloads" target="_blank" rel="noopener">这里</a>下载并安装git，看不懂就一直点next就行</p><p>git是一个命令行工具，要使用它只能敲对应的命令</p><p>下面介绍的命令均在对应目录下的命令行中执行</p><ul><li>如何在当前目录下打开命令行？<ul><li>如图在任务管理器路径前敲CMD即可</li></ul></li></ul><p><img src="/posts/简易版Git教程/1.png"></p><h2 id="git-clone">git clone</h2><p>clone是将网络上的仓库克隆到本地的操作</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git clone "https://github.com/linkfqy/EazyTime.git"</span><br></pre></td></tr></table></figure><p>上面的命令将我们项目的工程文件clone至当前目录</p><p>当前目录会生成一个EazyTime文件夹即工程文件夹，打开它并重新打开命令行，<strong>后续命令均在此文件夹中进行</strong></p><p>clone到本地后，你就可以对工程文件进行编辑了，可以任意增加想要的功能</p><h2 id="git-add">git add</h2><p>一般来说，我们只希望自己工作的那部分文件能够更新到仓库，而无关紧要的修改不应影响他人</p><p>因此需要对工作文件进行add</p><p>add命令将指定的文件或目录添加至<strong>本地</strong>的暂存区，只有add过的文件才能被提交，add过的文件还可以修改，也可以再次add，最终提交的文件以最后一次add的版本为准</p><p>如：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git add "README.md"</span><br></pre></td></tr></table></figure><p>以上命令将<code>README.md</code>文件添加至<strong>本地</strong>的暂存区</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git add "app\src\main\java"</span><br></pre></td></tr></table></figure><p>以上命令将<code>app\src\main\java</code>整个文件夹添加至<strong>本地</strong>的暂存区</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git add *</span><br></pre></td></tr></table></figure><p>以上命令将<strong>所有文件</strong>添加至本地的暂存区</p><p><strong>注意：不要随意使用以上命令！因为这会将你对工程做的所有修改（而非仅属于你的那部分工作）都添加！一旦push，会对所有人共用的仓库产生影响！</strong></p><p>合适的做法是，仅add自己工作所属的目录，如：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git add "app\src\main\java\com\hitsz\eazytime\ui\todo"</span><br></pre></td></tr></table></figure><h2 id="git-status">git status</h2><p><code>git status</code>命令用于显示工作目录和暂存区的状态。</p><p>使用此命令能看到哪些<strong>修改</strong>被暂存到了, 哪些没有，哪些<strong>文件</strong>还没被add。<code>git status</code>不显示已经<code>commit</code>到项目历史中去的信息。</p><p>“<em>Changes to be committed</em>”所列的是已add过的文件版本 “<em>Changed but not updated</em>”所列的是add过但被修改，尚未再次add的文件 “<em>Untracked files</em>”中所列的是尚未add的文件</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line">Changes to be committed:</span><br><span class="line"> (use "git restore --staged <file>..." to unstage)</span><br><span class="line"> modified: README.md</span><br><span class="line"></span><br><span class="line">Changes not staged for commit:</span><br><span class="line"> (use "git add <file>..." to update what will be committed)</span><br><span class="line"> (use "git restore <file>..." to discard changes in working directory)</span><br><span class="line"> modified: README.md</span><br></pre></td></tr></table></figure><p>以上表示<code>README.md</code>被add过一个版本，但之后又修改过，可以再次add以添加新的版本</p><p>同时也提示你：</p><ul><li>想要舍弃已add的版本，使用<code>git restore --staged "README.md"</code>命令</li><li>想要还原对<code>README.md</code>的修改，使用<code>git restore "README.md"</code>将文件变为最后一次add的版本</li></ul><h2 id="git-commit">git commit</h2><p><code>commit</code>命令将暂存区的内容添加到<strong>本地</strong>仓库中</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git commit -m "message"</span><br></pre></td></tr></table></figure><p>上面的message是对此次提交的注释，可以写点关于这次提交干了啥的解释</p><h2 id="git-push">git push</h2><p>终于到了最后一步，将本地所作的修改提交至GitHub上的仓库</p><p><strong>注意：之前的操作都是本地的，不push大家是看不到你的工作的</strong></p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git push</span><br></pre></td></tr></table></figure><p>一句话将本地仓库提交至GitHub</p><p>如果想要提交对应分支，则：</p><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">git push origin main</span><br></pre></td></tr></table></figure><p>将本地的main分支提交至GitHub</p><p>第一次push可能会弹出登录窗口，登录github账号就可以正常提交了</p>]]></content>
<summary type="html">
<h1 id="简易版git教程">简易版git教程</h1>
<p>这是写给大一立项小组成员看的简易版git教程</p>
<p>我们做这个项目只需要一些基本的git命令就可以了，因此这里教大家10min速成git</p>
<h2 id="准备工作">准备工作</h2>
</summary>
<category term="杂" scheme="http://oi.linkfqy.top/categories/%E6%9D%82/"/>
<category term="git" scheme="http://oi.linkfqy.top/tags/git/"/>
</entry>
<entry>
<title>EazyTime开发笔记</title>
<link href="http://oi.linkfqy.top/posts/EazyTime/"/>
<id>http://oi.linkfqy.top/posts/EazyTime/</id>
<published>2021-03-29T15:12:08.000Z</published>
<updated>2021-03-30T14:15:00.600Z</updated>
<content type="html"><![CDATA[]]></content>
<summary type="html">
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
</entry>
<entry>
<title>Python学习笔记</title>
<link href="http://oi.linkfqy.top/posts/Python%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<id>http://oi.linkfqy.top/posts/Python学习笔记/</id>
<published>2021-03-25T15:35:00.000Z</published>
<updated>2021-03-27T02:33:17.483Z</updated>
<content type="html"><![CDATA[<h2 id="section">3.25</h2><p>今天遇到了这样一个问题：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">ListNode</span>:</span></span><br><span class="line"> <span class="function"><span class="keyword">def</span> <span class="title">__init__</span><span class="params">(self, val=<span class="number">0</span>, next=None)</span>:</span></span><br><span class="line"> self.val = val</span><br><span class="line"> self.next = next</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">def</span> <span class="title">set</span><span class="params">(self, L)</span>:</span></span><br><span class="line"> self=ListNode(val=L[<span class="number">0</span>])</span><br><span class="line"> now = self</span><br><span class="line"> <span class="keyword">for</span> x <span class="keyword">in</span> L[<span class="number">1</span>:]:</span><br><span class="line"> now.next = ListNode(val=x)</span><br><span class="line"> now = now.next</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">def</span> <span class="title">print</span><span class="params">(self)</span>:</span></span><br><span class="line"> now = self</span><br><span class="line"> <span class="keyword">while</span> (now != <span class="keyword">None</span>):</span><br><span class="line"> print(<span class="string">'%d '</span> % (now.val))</span><br><span class="line"> now = now.next</span><br><span class="line"> print(<span class="string">'\n'</span>)</span><br><span class="line"> </span><br><span class="line">a=ListNode()</span><br><span class="line">a.set([<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>])</span><br><span class="line">a.print() <span class="comment"># 输出0</span></span><br></pre></td></tr></table></figure><p>可见，<code>a.set()</code>并没有改变a的值，不得其解</p><p>看了这篇<a href="https://www.cnblogs.com/pythonista/p/11178705.html" target="_blank" rel="noopener">文章</a></p><p>豁然开朗，之前理解有误的原因是自动把C++的赋值与Python混淆了</p><p>Python的变量，其实是C++意义上的指针</p><p>下面摘录几段：</p><hr><h3 id="构造函数返回指针">构造函数返回指针</h3><p><strong>Python 的构造函数将构造匿名对象，且返回此对象的一个指针。</strong></p><p>这是 Python 与指针的第一个重要联系。</p><p>用代码描述，对于Python代码：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">sampleNum = <span class="number">0</span></span><br></pre></td></tr></table></figure><p>其不类似于 C++ 代码：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">int</span> sampleNum = <span class="number">0</span>;</span><br></pre></td></tr></table></figure><p>而更类似于：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">int</span> __tmpNum = <span class="number">0</span>, *sampleNum = &__tmpNum;</span><br><span class="line"></span><br><span class="line"><span class="comment">// 或者：</span></span><br><span class="line"><span class="built_in">shared_ptr</span><<span class="keyword">int</span>> sampleNum(<span class="keyword">new</span> <span class="keyword">int</span>(<span class="number">0</span>));</span><br></pre></td></tr></table></figure><h3 id="setitems__操作将隐式解指针">__setitems__操作将隐式解指针</h3><p>Python 与指针的另一个重要联系在于 Python 的<strong>隐式解指针行为。</strong></p><p>虽然 Python 不存在显式解指针操作，但（有且仅有）__setitems__操作将进行隐式解指针，通过此方法对变量进行修改等同于通过解指针操作修改变量原值。</p><p>此种性质意味着：</p><ol type="1"><li>任何不涉及__setitems__的操作都将成为指针重绑定。</li></ol><p>对于Python代码：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">numList = [<span class="keyword">None</span>] * <span class="number">10</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># Rebinding</span></span><br><span class="line">numList = [<span class="keyword">None</span>] * <span class="number">5</span></span><br></pre></td></tr></table></figure><p>其相当于：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">int</span> *numList = <span class="keyword">new</span> <span class="keyword">int</span>[<span class="number">10</span>];</span><br><span class="line"></span><br><span class="line"><span class="comment">// Rebinding</span></span><br><span class="line"><span class="keyword">delete</span>[] numList;</span><br><span class="line">numList = <span class="keyword">new</span> <span class="keyword">int</span>[<span class="number">5</span>];</span><br><span class="line"><span class="keyword">delete</span>[] numList;</span><br></pre></td></tr></table></figure><p>由此可见，对 numList 的非__setitems__操作，导致 numList 被绑定到了一个新指针上。</p><ol start="2" type="1"><li>任何涉及__setitems__的操作都将成为解指针操作。</li></ol><p>由于 Python 对哈希表的高度依赖，“涉及__setitems__的操作”在 Python 中实际上是一个非常广泛的行为，这主要包括：</p><ul><li>对数组的索引操作</li><li>对哈希表的查找操作</li><li>涉及__setattr__的操作（由于 Python 将 attribute 存储在哈希表中，所以__setattr__操作最终将是某种__setitems__操作）</li></ul><hr><p>于是，将<code>set()</code>改为这样：</p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">set</span><span class="params">(self, L)</span>:</span></span><br><span class="line"> self.val = L[<span class="number">0</span>] <span class="comment"># self=ListNode(val=L[0])</span></span><br><span class="line"> self.next = <span class="keyword">None</span></span><br><span class="line"> now = self</span><br><span class="line"> <span class="keyword">for</span> x <span class="keyword">in</span> L[<span class="number">1</span>:]:</span><br><span class="line"> now.next = ListNode(val=x)</span><br><span class="line"> now = now.next</span><br></pre></td></tr></table></figure><p>就解决了问题</p><p><strong><code>self</code>是默认将实例本身传入的参数</strong>，根据上文，实际传入的是实例的指针，而<strong><code>self=ListNode(val=L[0])</code>将<code>self</code>重新绑定至一个新建的节点</strong>，那么后续进行的操作就与原实例无关了</p>]]></content>
<summary type="html">
<h2 id="section">3.25</h2>
<p>今天遇到了这样一个问题：</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</s
</summary>
<category term="学习笔记" scheme="http://oi.linkfqy.top/categories/%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/"/>
<category term="Python" scheme="http://oi.linkfqy.top/tags/Python/"/>
</entry>
<entry>
<title>【最大权闭合子图】LOJ6045 「雅礼集训 2017 Day8」价</title>
<link href="http://oi.linkfqy.top/posts/LOJ6045/"/>
<id>http://oi.linkfqy.top/posts/LOJ6045/</id>
<published>2018-11-25T11:48:02.000Z</published>
<updated>2018-11-25T12:09:24.000Z</updated>
<content type="html"><![CDATA[<p><a href="https://loj.ac/problem/6045" target="_blank" rel="noopener">题面在这里</a></p><p></p><p>很明显是最大权闭合子图的模型</p><p></p><p>示例程序：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstring></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> cl(x,y) memset(x,y,sizeof(x))</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> ll;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">605</span>,maxe=<span class="number">200005</span>,inf=<span class="number">0x3f3f3f3f</span>,INF=<span class="number">0x7fffffff</span>;</span><br><span class="line"><span class="keyword">int</span> n,S,T;</span><br><span class="line"><span class="keyword">int</span> tot,son[maxe],nxt[maxe],lnk[maxn];ll flw[maxe],cap[maxe];</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y,<span class="keyword">int</span> f)</span></span>{</span><br><span class="line">son[++tot]=y;nxt[tot]=lnk[x];lnk[x]=tot;flw[tot]=<span class="number">0</span>;cap[tot]=f;</span><br><span class="line">son[++tot]=x;nxt[tot]=lnk[y];lnk[y]=tot;flw[tot]=<span class="number">0</span>;cap[tot]=<span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">int</span> que[maxn],d[maxn],pos[maxn];</span><br><span class="line"><span class="function"><span class="keyword">bool</span> <span class="title">bfs</span><span class="params">()</span></span>{</span><br><span class="line">cl(d,<span class="number">63</span>);</span><br><span class="line"><span class="keyword">int</span> hed=<span class="number">0</span>,til=<span class="number">1</span>;</span><br><span class="line">que[<span class="number">1</span>]=S;d[S]=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">while</span> (hed!=til){</span><br><span class="line"><span class="keyword">int</span> x=que[++hed];</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j=lnk[x];j;j=nxt[j])</span><br><span class="line"> <span class="keyword">if</span> (d[son[j]]==inf&&flw[j]<cap[j])</span><br><span class="line"> d[que[++til]=son[j]]=d[x]+<span class="number">1</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> d[T]!=inf;</span><br><span class="line">}</span><br><span class="line"><span class="function">ll <span class="title">dfs</span><span class="params">(<span class="keyword">int</span> x,ll flow)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (x==T||flow==<span class="number">0</span>) <span class="keyword">return</span> flow;</span><br><span class="line">ll f=<span class="number">0</span>,res=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> &j=pos[x];j;j=nxt[j])</span><br><span class="line"> <span class="keyword">if</span> (d[son[j]]==d[x]+<span class="number">1</span>&&(f=dfs(son[j],min(flow,cap[j]-flw[j])))><span class="number">0</span>){</span><br><span class="line"> flw[j]+=f;flw[j^<span class="number">1</span>]-=f;</span><br><span class="line"> res+=f,flow-=f;</span><br><span class="line"> <span class="keyword">if</span> (flow==<span class="number">0</span>) <span class="keyword">return</span> res;</span><br><span class="line"> }</span><br><span class="line"><span class="keyword">return</span> res;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d"</span>,&n);</span><br><span class="line">S=<span class="number">2</span>*n+<span class="number">1</span>,T=S+<span class="number">1</span>;tot=<span class="number">1</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++){</span><br><span class="line"><span class="keyword">int</span> t;<span class="built_in">scanf</span>(<span class="string">"%d"</span>,&t);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>,x;j<=t;j++) <span class="built_in">scanf</span>(<span class="string">"%d"</span>,&x),add(i,x+n,INF);</span><br><span class="line">add(i+n,T,inf);</span><br><span class="line">}</span><br><span class="line">ll ans=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>,pi;i<=n;i++) <span class="built_in">scanf</span>(<span class="string">"%d"</span>,&pi),add(S,i,-pi+inf),ans-=inf-pi;</span><br><span class="line"><span class="keyword">while</span> (bfs()){</span><br><span class="line"><span class="built_in">memcpy</span>(pos,lnk,<span class="keyword">sizeof</span>(lnk));</span><br><span class="line">ans+=dfs(S,INF);</span><br><span class="line">}</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%lld"</span>,ans);</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://loj.ac/problem/6045" target="_blank" rel="noopener">题面在这里</a></p>
<p></p>
<p>很明显是最大权闭合子图的模型</p>
<p></p>
<p>示例程序：</
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="二分图" scheme="http://oi.linkfqy.top/tags/%E4%BA%8C%E5%88%86%E5%9B%BE/"/>
<category term="最大权闭合子图" scheme="http://oi.linkfqy.top/tags/%E6%9C%80%E5%A4%A7%E6%9D%83%E9%97%AD%E5%90%88%E5%AD%90%E5%9B%BE/"/>
<category term="LOJ" scheme="http://oi.linkfqy.top/tags/LOJ/"/>
</entry>
<entry>
<title>【莫队+bitset】BZOJ4810 [Ynoi2017]由乃的玉米田</title>
<link href="http://oi.linkfqy.top/posts/BZOJ4810/"/>
<id>http://oi.linkfqy.top/posts/BZOJ4810/</id>
<published>2018-11-22T22:52:00.000Z</published>
<updated>2018-11-22T23:03:06.000Z</updated>
<content type="html"><![CDATA[<p><a href="https://www.lydsy.com/JudgeOnline/problem.php?id=4810" target="_blank" rel="noopener">题面在这里</a></p><p></p><p>直接莫队……用bitset维护出现过哪些值 <span class="math display">\[a-b=x\Leftrightarrow a=b+x \\a+b=x\Leftrightarrow a=-(b-x)\]</span></p><p>对于操作1，bitset左移x位后对自己取并</p><p>对于操作2，需要同时维护一个所有值的相反数的bitset，同1</p><p>对于操作3，直接爆枚因子即可</p><p>时间复杂度<span class="math inline">\(O(n\sqrt n)\)</span></p><p></p><p>示例程序：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><bitset></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cmath></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">char</span> <span class="title">nc</span><span class="params">()</span></span>{</span><br><span class="line"><span class="keyword">static</span> <span class="keyword">char</span> buf[<span class="number">100000</span>],*l=buf,*r=buf;</span><br><span class="line"><span class="keyword">return</span> l==r&&(r=(l=buf)+fread(buf,<span class="number">1</span>,<span class="number">100000</span>,<span class="built_in">stdin</span>),l==r)?EOF:*l++;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">red</span><span class="params">()</span></span>{</span><br><span class="line"><span class="keyword">int</span> res=<span class="number">0</span>,f=<span class="number">1</span>;<span class="keyword">char</span> ch=nc();</span><br><span class="line"><span class="keyword">while</span> (ch<<span class="string">'0'</span>||<span class="string">'9'</span><ch) {<span class="keyword">if</span> (ch==<span class="string">'-'</span>) f=-f;ch=nc();}</span><br><span class="line"><span class="keyword">while</span> (<span class="string">'0'</span><=ch&&ch<=<span class="string">'9'</span>) res=res*<span class="number">10</span>+ch<span class="number">-48</span>,ch=nc();</span><br><span class="line"><span class="keyword">return</span> res*f;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">100005</span>;</span><br><span class="line"><span class="keyword">int</span> n,q,a[maxn],h[maxn];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Query</span>{</span></span><br><span class="line"><span class="keyword">int</span> t,l,r,x,id;</span><br><span class="line"><span class="keyword">bool</span> <span class="keyword">operator</span><(<span class="keyword">const</span> Query&b)<span class="keyword">const</span>{</span><br><span class="line"><span class="keyword">if</span> (h[l]==h[b.l]) <span class="keyword">return</span> r<b.r;</span><br><span class="line"><span class="keyword">return</span> l<b.l;</span><br><span class="line">}</span><br><span class="line">}Q[maxn];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">blocker</span><span class="params">()</span></span>{</span><br><span class="line"><span class="keyword">int</span> k=<span class="built_in">sqrt</span>(n);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++) h[i]=i/k;</span><br><span class="line">}</span><br><span class="line"><span class="built_in">bitset</span><maxn*2> A,B;</span><br><span class="line"><span class="keyword">int</span> cnt[maxn*<span class="number">2</span>];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> x)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (!cnt[x]) A[x+<span class="number">100000</span>]=<span class="number">1</span>;</span><br><span class="line"><span class="keyword">if</span> (!cnt[x]) B[<span class="number">100000</span>-x]=<span class="number">1</span>;</span><br><span class="line">cnt[x]++;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">dec</span><span class="params">(<span class="keyword">int</span> x)</span></span>{</span><br><span class="line">cnt[x]--;</span><br><span class="line"><span class="keyword">if</span> (!cnt[x]) A[x+<span class="number">100000</span>]=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">if</span> (!cnt[x]) B[<span class="number">100000</span>-x]=<span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">bool</span> ans[maxn];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line">n=red(),q=red();</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++) a[i]=red();</span><br><span class="line">blocker();</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=q;i++) Q[i].t=red(),Q[i].l=red(),Q[i].r=red(),Q[i].x=red(),Q[i].id=i;</span><br><span class="line">sort(Q+<span class="number">1</span>,Q+<span class="number">1</span>+q);</span><br><span class="line"><span class="keyword">int</span> L=<span class="number">1</span>,R=<span class="number">1</span>; add(a[<span class="number">1</span>]);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=q;i++){</span><br><span class="line"><span class="keyword">while</span> (Q[i].l<L) add(a[--L]);</span><br><span class="line"><span class="keyword">while</span> (R<Q[i].r) add(a[++R]);</span><br><span class="line"><span class="keyword">while</span> (L<Q[i].l) dec(a[L++]);</span><br><span class="line"><span class="keyword">while</span> (Q[i].r<R) dec(a[R--]);</span><br><span class="line"><span class="keyword">if</span> (Q[i].t==<span class="number">1</span>){</span><br><span class="line">ans[Q[i].id]=(A&(A<<Q[i].x)).any();</span><br><span class="line">}<span class="keyword">else</span></span><br><span class="line"><span class="keyword">if</span> (Q[i].t==<span class="number">2</span>){</span><br><span class="line">ans[Q[i].id]=(A&(B<<Q[i].x)).any();</span><br><span class="line">}<span class="keyword">else</span>{</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> d=<span class="number">1</span>,td=<span class="built_in">sqrt</span>(Q[i].x);d<=td;d++)</span><br><span class="line"> <span class="keyword">if</span> (Q[i].x%d==<span class="number">0</span>&&cnt[Q[i].x/d]&&cnt[d]) {ans[Q[i].id]=<span class="number">1</span>;<span class="keyword">break</span>;}</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=q;i++) <span class="built_in">puts</span>(ans[i]?<span class="string">"yuno"</span>:<span class="string">"yumi"</span>);</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://www.lydsy.com/JudgeOnline/problem.php?id=4810" target="_blank" rel="noopener">题面在这里</a></p>
<p></p>
<p>直接莫队……用bitset维
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="BZOJ" scheme="http://oi.linkfqy.top/tags/BZOJ/"/>
<category term="莫队" scheme="http://oi.linkfqy.top/tags/%E8%8E%AB%E9%98%9F/"/>
</entry>
<entry>
<title>【二分图博弈】LOJ6033 「雅礼集训 2017 Day2」棋盘游戏</title>
<link href="http://oi.linkfqy.top/posts/LOJ6033/"/>
<id>http://oi.linkfqy.top/posts/LOJ6033/</id>
<published>2018-11-22T09:37:58.000Z</published>
<updated>2018-11-22T10:01:44.000Z</updated>
<content type="html"><![CDATA[<p><a href="https://loj.ac/problem/6033" target="_blank" rel="noopener">题面在这里</a></p><p></p><p>显然可以把整个图黑白染色，得到一个二分图</p><p>考虑二分图最大匹配，如果初始点是未匹配点，则Bob只能走未匹配边到一个匹配点，而Alice一定可以沿匹配边走回左边的匹配点，由于不能往回走，此时的情形与初始状态一样，最终Bob输</p><p>如果初始点是匹配点，此时Bob走的是匹配边，Alice走未匹配边，如果最后能走回左边则Alice赢，否则Bob赢。其实他们走的是增广路，Alice能赢当且仅当初始点不一定是匹配点</p><p>所以答案是所有【不一定是匹配点】的点</p><p>具体做法是从左边未匹配点出发，沿增广路遍历到的所有左边点都是答案</p><p></p><p>示例程序：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstring></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> cl(x,y) memset(x,y,sizeof(x))</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">10005</span>,maxe=<span class="number">40005</span>,p[<span class="number">4</span>][<span class="number">2</span>]={{<span class="number">0</span>,<span class="number">1</span>},{<span class="number">1</span>,<span class="number">0</span>},{<span class="number">0</span>,<span class="number">-1</span>},{<span class="number">-1</span>,<span class="number">0</span>}};</span><br><span class="line"><span class="keyword">int</span> n,m,N,id[<span class="number">105</span>][<span class="number">105</span>];</span><br><span class="line"><span class="keyword">char</span> a[<span class="number">105</span>][<span class="number">105</span>];</span><br><span class="line"><span class="keyword">int</span> tot,son[maxe],lnk[maxn],nxt[maxe];</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>{</span><br><span class="line">son[++tot]=y;nxt[tot]=lnk[x];lnk[x]=tot;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">int</span> vis[maxn],times,con[maxn];</span><br><span class="line"><span class="function"><span class="keyword">bool</span> <span class="title">fnd</span><span class="params">(<span class="keyword">int</span> x)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (vis[x]==times) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">vis[x]=times;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j=lnk[x];j;j=nxt[j]){</span><br><span class="line"><span class="keyword">int</span> k=con[son[j]];con[son[j]]=x;</span><br><span class="line"><span class="keyword">if</span> (!k||fnd(k)) <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line">con[son[j]]=k;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">bool</span> ans[maxn][maxn];</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">dfs</span><span class="params">(<span class="keyword">int</span> x)</span></span>{</span><br><span class="line">vis[x]=<span class="number">1</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j=lnk[x];j;j=nxt[j])</span><br><span class="line"> <span class="keyword">if</span> (!vis[con[son[j]]]) dfs(con[son[j]]);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&n,&m);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++) <span class="built_in">scanf</span>(<span class="string">"%s"</span>,a[i]+<span class="number">1</span>);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++)</span><br><span class="line"> <span class="keyword">if</span> (a[i][j]==<span class="string">'.'</span>) id[i][j]=++N;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++) <span class="keyword">if</span> (a[i][j]==<span class="string">'.'</span>)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> t=<span class="number">0</span>;t<<span class="number">4</span>;t++){</span><br><span class="line"> <span class="keyword">int</span> ii=i+p[t][<span class="number">0</span>],jj=j+p[t][<span class="number">1</span>];</span><br><span class="line"> <span class="keyword">if</span> (ii<<span class="number">1</span>||jj<<span class="number">1</span>||ii>n||jj>m||a[ii][jj]==<span class="string">'#'</span>) <span class="keyword">continue</span>;</span><br><span class="line"> add(id[i][j],id[ii][jj]);</span><br><span class="line"> }</span><br><span class="line"><span class="keyword">int</span> num=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++)</span><br><span class="line"> <span class="keyword">if</span> (a[i][j]==<span class="string">'.'</span>&&((i+j)&<span class="number">1</span>)) times++,num+=fnd(id[i][j]);</span><br><span class="line"><span class="keyword">if</span> (num*<span class="number">2</span>==N) <span class="keyword">return</span> <span class="built_in">puts</span>(<span class="string">"0"</span>),<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=N;i++) con[con[i]]=i;</span><br><span class="line"></span><br><span class="line">cl(vis,<span class="number">0</span>);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++)</span><br><span class="line"> <span class="keyword">if</span> (a[i][j]==<span class="string">'.'</span>&&!vis[id[i][j]])</span><br><span class="line"> <span class="keyword">if</span> (!con[id[i][j]]) dfs(id[i][j]);</span><br><span class="line"><span class="keyword">int</span> ANS=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++)</span><br><span class="line"> <span class="keyword">if</span> (vis[id[i][j]]) ANS++,ans[i][j]=<span class="number">1</span>;</span><br><span class="line"></span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%d\n"</span>,ANS);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=m;j++)</span><br><span class="line"> <span class="keyword">if</span> (ans[i][j]) <span class="built_in">printf</span>(<span class="string">"%d %d\n"</span>,i,j);</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://loj.ac/problem/6033" target="_blank" rel="noopener">题面在这里</a></p>
<p></p>
<p>显然可以把整个图黑白染色，得到一个二分图</p>
<p>考虑二分图最大匹配，如
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="二分图" scheme="http://oi.linkfqy.top/tags/%E4%BA%8C%E5%88%86%E5%9B%BE/"/>
<category term="博弈论" scheme="http://oi.linkfqy.top/tags/%E5%8D%9A%E5%BC%88%E8%AE%BA/"/>
<category term="LOJ" scheme="http://oi.linkfqy.top/tags/LOJ/"/>
</entry>
<entry>
<title>LOJ6030 「雅礼集训 2017 Day1」矩阵</title>
<link href="http://oi.linkfqy.top/posts/LOJ6030/"/>
<id>http://oi.linkfqy.top/posts/LOJ6030/</id>
<published>2018-11-16T12:47:02.000Z</published>
<updated>2018-11-16T12:48:42.000Z</updated>
<content type="html"><![CDATA[<p><a href="https://loj.ac/problem/6030" target="_blank" rel="noopener">题面在这里</a></p><p></p><p>直接乱搞即可</p><p></p><p>示例程序：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">1005</span>;</span><br><span class="line"><span class="keyword">int</span> n,x[maxn],y[maxn];</span><br><span class="line"><span class="keyword">char</span> s[maxn];</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d"</span>,&n);</span><br><span class="line"><span class="keyword">bool</span> suc=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++){</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%s"</span>,s+<span class="number">1</span>);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> j=<span class="number">1</span>;j<=n;j++)</span><br><span class="line"> <span class="keyword">if</span> (s[j]==<span class="string">'#'</span>) suc=<span class="number">1</span>,x[i]++,y[j]++;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">if</span> (!suc) <span class="keyword">return</span> <span class="built_in">puts</span>(<span class="string">"-1"</span>),<span class="number">0</span>;</span><br><span class="line"><span class="keyword">int</span> ans=n*n;</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++) ans=min(ans,n-x[i]+!y[i]);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++)</span><br><span class="line"> <span class="keyword">if</span> (y[i]!=n) ans++;</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%d"</span>,ans);</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://loj.ac/problem/6030" target="_blank" rel="noopener">题面在这里</a></p>
<p></p>
<p>直接乱搞即可</p>
<p></p>
<p>示例程序：</p>
<fig
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="贪心" scheme="http://oi.linkfqy.top/tags/%E8%B4%AA%E5%BF%83/"/>
<category term="LOJ" scheme="http://oi.linkfqy.top/tags/LOJ/"/>
</entry>
<entry>
<title>【线段树+势能分析】LOJ6029 「雅礼集训 2017 Day1」市场</title>
<link href="http://oi.linkfqy.top/posts/LOJ6029/"/>
<id>http://oi.linkfqy.top/posts/LOJ6029/</id>
<published>2018-11-16T09:52:29.000Z</published>
<updated>2018-11-16T10:09:08.000Z</updated>
<content type="html"><![CDATA[<p><a href="https://loj.ac/problem/6029" target="_blank" rel="noopener">题面在这里</a></p><p></p><p>考虑一个区间除后变化量如果都相同，就转化为区间减了</p><p>维护最大值和最小值即可判断</p><p>复杂度证明：</p><p>考虑两个数<span class="math inline">\(a,b\)</span>，差是<span class="math inline">\(a-b\)</span>，除以<span class="math inline">\(d\)</span>后差是<span class="math inline">\(\frac{a-b}d\)</span></p><p>也就是说两个数被除<span class="math inline">\(\log\)</span>次就相等了</p><p>令一个线段树节点的势能为<span class="math inline">\(\log(Max-Min)\)</span></p><p>操作2就相当于把区间内所有势能不为1的节点势能-1，代价为势能减小总量</p><p>操作1会将<span class="math inline">\(\log\)</span>个节点的势能恢复为<span class="math inline">\(\log(Max-Min)\)</span></p><p>总时间复杂度为整个过程中产生的势能总量<span class="math inline">\(O((n+q\log n)\log a_i)\)</span></p><p></p><p>示例程序：</p><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cstdio></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><cmath></span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string"><algorithm></span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> ll;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> maxn=<span class="number">100005</span>,maxs=<span class="number">400005</span>;</span><br><span class="line"><span class="keyword">int</span> n,q,a[maxn];</span><br><span class="line">ll mx[maxs],mn[maxs],s[maxs],ad[maxs],len[maxs];</span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> ls x<<1</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> rs x<<1|1</span></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">pushup</span><span class="params">(<span class="keyword">int</span> x)</span> </span>{s[x]=s[ls]+s[rs];mx[x]=max(mx[ls],mx[rs]);mn[x]=min(mn[ls],mn[rs]);}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">addad</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> w)</span> </span>{mx[x]+=w;mn[x]+=w;ad[x]+=w;s[x]+=len[x]*w;}</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">pushdown</span><span class="params">(<span class="keyword">int</span> x)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (ad[x]) addad(ls,ad[x]),addad(rs,ad[x]),ad[x]=<span class="number">0</span>;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">build</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span>{</span><br><span class="line">ad[x]=<span class="number">0</span>; len[x]=r-l+<span class="number">1</span>;</span><br><span class="line"><span class="keyword">if</span> (l==r) <span class="keyword">return</span> mx[x]=mn[x]=s[x]=a[l],<span class="keyword">void</span>();</span><br><span class="line"><span class="keyword">int</span> mid=l+r>><span class="number">1</span>;</span><br><span class="line">build(ls,l,mid);build(rs,mid+<span class="number">1</span>,r);</span><br><span class="line">pushup(x);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">istad</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> ql,<span class="keyword">int</span> qr,<span class="keyword">int</span> w)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (qr<l||r<ql) <span class="keyword">return</span>;</span><br><span class="line"><span class="keyword">if</span> (ql<=l&&r<=qr) <span class="keyword">return</span> addad(x,w);</span><br><span class="line"><span class="keyword">int</span> mid=l+r>><span class="number">1</span>; pushdown(x);</span><br><span class="line">istad(ls,l,mid,ql,qr,w);istad(rs,mid+<span class="number">1</span>,r,ql,qr,w);</span><br><span class="line">pushup(x);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">istdv</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> ql,<span class="keyword">int</span> qr,<span class="keyword">int</span> d)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (qr<l||r<ql) <span class="keyword">return</span>;</span><br><span class="line"><span class="keyword">if</span> (ql<=l&&r<=qr&&<span class="built_in">floor</span>(<span class="number">1.0</span>*mx[x]/d)-mx[x]==<span class="built_in">floor</span>(<span class="number">1.0</span>*mn[x]/d)-mn[x]) <span class="keyword">return</span> addad(x,<span class="built_in">floor</span>(<span class="number">1.0</span>*mn[x]/d)-mn[x]);</span><br><span class="line"><span class="keyword">int</span> mid=l+r>><span class="number">1</span>; pushdown(x);</span><br><span class="line">istdv(ls,l,mid,ql,qr,d);istdv(rs,mid+<span class="number">1</span>,r,ql,qr,d);</span><br><span class="line">pushup(x);</span><br><span class="line">}</span><br><span class="line"><span class="function">ll <span class="title">qrymn</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> ql,<span class="keyword">int</span> qr)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (qr<l||r<ql) <span class="keyword">return</span> <span class="number">1e18</span>;</span><br><span class="line"><span class="keyword">if</span> (ql<=l&&r<=qr) <span class="keyword">return</span> mn[x];</span><br><span class="line"><span class="keyword">int</span> mid=l+r>><span class="number">1</span>; pushdown(x);</span><br><span class="line"><span class="keyword">return</span> min(qrymn(ls,l,mid,ql,qr),qrymn(rs,mid+<span class="number">1</span>,r,ql,qr));</span><br><span class="line">}</span><br><span class="line"><span class="function">ll <span class="title">qrys</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> ql,<span class="keyword">int</span> qr)</span></span>{</span><br><span class="line"><span class="keyword">if</span> (qr<l||r<ql) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"><span class="keyword">if</span> (ql<=l&&r<=qr) <span class="keyword">return</span> s[x];</span><br><span class="line"><span class="keyword">int</span> mid=l+r>><span class="number">1</span>; pushdown(x);</span><br><span class="line"><span class="keyword">return</span> qrys(ls,l,mid,ql,qr)+qrys(rs,mid+<span class="number">1</span>,r,ql,qr);</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"><span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&n,&q);</span><br><span class="line"><span class="keyword">for</span> (<span class="keyword">int</span> i=<span class="number">1</span>;i<=n;i++) <span class="built_in">scanf</span>(<span class="string">"%d"</span>,&a[i]);</span><br><span class="line">build(<span class="number">1</span>,<span class="number">1</span>,n);</span><br><span class="line"><span class="keyword">while</span> (q--){</span><br><span class="line"><span class="keyword">int</span> c;<span class="built_in">scanf</span>(<span class="string">"%d"</span>,&c);</span><br><span class="line"><span class="keyword">if</span> (c==<span class="number">1</span>){</span><br><span class="line"><span class="keyword">int</span> l,r,w;<span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>,&l,&r,&w);l++,r++;</span><br><span class="line">istad(<span class="number">1</span>,<span class="number">1</span>,n,l,r,w);</span><br><span class="line">}<span class="keyword">else</span></span><br><span class="line"><span class="keyword">if</span> (c==<span class="number">2</span>){</span><br><span class="line"><span class="keyword">int</span> l,r,w;<span class="built_in">scanf</span>(<span class="string">"%d%d%d"</span>,&l,&r,&w);l++,r++;</span><br><span class="line">istdv(<span class="number">1</span>,<span class="number">1</span>,n,l,r,w);</span><br><span class="line">}<span class="keyword">else</span></span><br><span class="line"><span class="keyword">if</span> (c==<span class="number">3</span>){</span><br><span class="line"><span class="keyword">int</span> l,r;<span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&l,&r);l++,r++;</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%lld\n"</span>,qrymn(<span class="number">1</span>,<span class="number">1</span>,n,l,r));</span><br><span class="line">}<span class="keyword">else</span>{</span><br><span class="line"><span class="keyword">int</span> l,r;<span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&l,&r);l++,r++;</span><br><span class="line"><span class="built_in">printf</span>(<span class="string">"%lld\n"</span>,qrys(<span class="number">1</span>,<span class="number">1</span>,n,l,r));</span><br><span class="line">}</span><br><span class="line">}</span><br><span class="line"><span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html">
<p><a href="https://loj.ac/problem/6029" target="_blank" rel="noopener">题面在这里</a></p>
<p></p>
<p>考虑一个区间除后变化量如果都相同，就转化为区间减了</p>
<p>维护最大值和
</summary>
<category term="题解" scheme="http://oi.linkfqy.top/categories/%E9%A2%98%E8%A7%A3/"/>
<category term="复杂度分析" scheme="http://oi.linkfqy.top/tags/%E5%A4%8D%E6%9D%82%E5%BA%A6%E5%88%86%E6%9E%90/"/>
<category term="线段树" scheme="http://oi.linkfqy.top/tags/%E7%BA%BF%E6%AE%B5%E6%A0%91/"/>
<category term="LOJ" scheme="http://oi.linkfqy.top/tags/LOJ/"/>
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